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in this video we're going to go over complex numbers we're going to talk about how to graph them how to calculate the absolute value we're going to work on problems on adding subtracting multiplying dividing complex numbers and even solving equations let's begin so typically you have something known as standard form which is written as a plus bi a represents the real portion of the standard or complex number so a is a real number B is the the term bi is the imaginary part of the number so let's say if we have the number 3 plus 4i how can we graph this number and also how can we calculate the absolute value of 3 plus 4 so let's make a graph the x-axis is the real axis the y-axis is known as the imaginary axis so the real part of the number is string so you want to travel 3 units to the right the imaginary part is 4 so you want to go up 4 units so the point 3 plus 4i lies right here now to calculate the absolute value of a plus bi it's equal to the square root of a squared plus B squared so basically it represents the hypotenuse of this triangle so what is 3 squared plus 4 squared 3 squared is 9 4 squared is 16 if you add them you get 25 which is 5 so that's the hypotenuse of the right triangle that's form so the absolute value of 3 plus 4i is 5 now let's try another example go ahead and plot this number negative 5 plus 12i and also go ahead and calculate the absolute value at the same time so this time we need to travel negative five units to the left that's on the real axis and on the imaginary axis we need to travel 12 units up so let's say it's somewhere in this vicinity so that's the point that we need to plot and let's make a right triangle out of it so we went five units left 12 units up and let's calculate the hypotenuse formed by this right triangle so the absolute value of a plus bi is equal to the square root of a squared which is negative five squared plus B squared which is 12 squared negative five squared is positive 25 and 12 squared is 144 if you had these two numbers you should get 169 and the square root of 169 is 13 so that's the hypotenuse of the triangle so the absolute value of negative 5 plus 12 is 13 and this is where you plot it try this one go ahead and plot eight negative 59 and find the absolute value so here's the real axis the imaginary y-axis and we need to travel eight units to the right and down 15 minutes so this time we're in quadrant four this is quadrant one two three and here's the fourth quadrant so the point of interest is right here that's how you plot it but let's turn it into a triangle so let's travel eight units to the right and down 15 units let's find the hypotenuse of this triangle so the absolute value of a plus bi is equal to the square root of 8 squared plus 15 squared 8 times 8 is 64 and 15 times 15 is 225 negative 15 times negative 15 is also positive 225 if we add 64 and 225 that's going to give us 289 and the square root of 289 is 17 so the absolute value is 17 that's the length of the hypotenuse this is basically the 8 15 17 triangle it helps familias rush your triangles so far we went over three of them there is the 3 4 5 triangle 3 squared plus 4 squared equals 5 squared we talked about the 5 12 13 triangle there's the 8 15 17 triangle there's also the 7 24 25 triangle these are the four most common right triangles that you'll encounter you may also see any ratios of the special triangles for example 6 8 10 works as well because that's a ratio or multiple of the 3 4 5 triangle 10 24 26 also works and there's some other ones like 9 40 41 and 11 60 61 now if you were asked to simplify these numbers what would you do let's say if you have two numbers to simplify the square root of 4 and the square root of negative 4 the square root of 4 will give you a real number in this case 2 because 2 times 2 is 4 but what about the square root of negative 4 the square root of negative 4 is 2i it's not just to the imaginary number I is equal to negative 1 so the square root of negative 4 is an imaginary number anytime you have a negative inside an even radical or radical with an even index number you're going to get an imaginary number so go ahead and simplify these two numbers the square root of 9 and the square root of negative 9 the square root of 9 is simply 3 the square root of negative 9 is the square root of 9 times the square root of negative 1 the square root of 9 is 3 and the square root of negative 1 is I so you get 3 I simplify these numbers square root of 25 square root negative 25 negative square root of 25 and negative square root of negative 25 so the square root of 25 is simply positive 5 the square root of negative 25 is positive 5i negative square root of 25 is going to be negative 5 and the last one is going to be negative 5 on now what about simplifying the square root of negative 18 the square root of 18 is not a perfect square so what would you do in a situation like this well it helps to know what the perfect squares are 1 squared is 1 2 squared is 4 3 squared is 9 4 squared is 16 5 squared is 25 6 squared is 36 7 squared is 49 all of these numbers are perfect squares you can take the square root of 9 and get a whole number 3 so you want to find out what perfect square goes into 18 the perfect square that goes into it is 9 18 is 9 times 2 and let's not forget the negative sign so we'll take out the negative 1 this word root of 9 is 3 we can't take the square root of 2 because it's not a perfect square and the square root of negative 1 is I so the final answer is 3 radical 2 I let's try another example like that go ahead and simplify a negative square root negative 50 so this is negative times square root 25 times square root 2 times square root of negative 1 now 25 is a perfect square the square root of 25 is 5 and the square root of negative 1 is I so this is the answer negative 5 radical 2 I try this one what is the square root of negative 80 16 is the highest perfect square that goes into 80 16 times 5 is 80 and the square root of 16 is 4 so the answer is 4 root 5 times I here is one more example for you simplify negative square root negative 72 so what perfect square goes into 72 9 goes into 72 that's a perfect square but 36 goes into 72 and 36 is larger than 9 so we want to write this as 36 times 2 72 divided by 36 is 2 that so I could find this missing number and let's not forget the square root of negative 1 the square root of 36 is 6 and the square root of negative 1 is I so the final answer is negative 6 radical 2 I now let's say if you were to get a question that looks like this what is I races to the 201 what is that what is that even equal how would you simplify that before we can do that problem you need to understand a few things so we know that I is equal to the square root of negative 1 now what about I squared and what about I to the 3rd and I to the 4th power you need to know what these values are equal to as well I squared is equal to negative 1 if you square the square root of negative 1 the 2 and the radical will disappear giving you negative 1 I to the 3rd is basically I squared times I because 2 plus 1 is 3 I squared is negative 1 so this is negative 1 times I or basically negative I so I to the third is equal to negative I and I squared is negative 1 the last one is I to the 4th which is I squared times I squared and that's a negative 1 times a negative 1 which is positive 1 make sure you know these 4 values now let's say if you want to simplify I to the sixth power what would you do I to the sixth power is basically I to the fourth power times I squared the reason why you want to break it up this way is because you know that I to the fourth is equal to one and I squared is negative one so therefore I to the sixth is simply negative one so let's try another example let's see if we want to simplify I raised to the 8th power now notice that 8 is a multiple of four so you can write it as I to the fourth times I to the fourth or simply I to the fourth raised to the second power since you have two of them and four times two is eight whenever you write it like this you need to add the exponents four plus four is eight if you raise one exponent to another you need to multiply four times two is eight now we know I to the fourth is 1 and 1 squared is 1 so if you have a I number with an exponent that is a multiple of eight it will always equal to one so for example I to the twelfth this is one as well it's I to the fourth raised to the third power since 4 times 3 is 12 and 1 to the third power is still one now what about let's say I to the 15 power how can we simplify this number so I to the 15 is I to the 12th times I to the third so one of these numbers you want it to be the highest multiple of four that's closest to 15 multiples of four are 4 8 12 and 16 16 is too much so you want to choose 12 and 15 minus 12 gives you 3 so you want to break it up this way and I to the 12 is I to the fourth raised to the third power which is 1 to the third power and we know that IQ is equal to negative I so the final answer is 1 times a negative I or simply negative I now for those of you who may want to know another way or systematic way of getting these two numbers especially if this number becomes very large here's what you could do take 15 and divide it by 4 if you type this in your calculator you should get 3.75 now take the whole number portion of this decimal number and multiply by 4 3 times 4 is 12 that gives you this number now the decimal portion of the number and multiply that by 8 4 so point 7 5 times 4 gives you 3 which is the other remaining number that's how you can find these two numbers if you ever have difficulty seeing what it is let's try another example simplify this one I to the 29th so this is I to the 28th times I and I to the 28th is basically I to the fourth raised to the seventh since 4 times 7 is 28 and this is going to be I to the 4th is 1 and 1 to the 7th power is 1 so 1 times I is simply on so I to the 29th is the same as I'm so one way to get these numbers 28 and one say 29 divided by 4 29 divided by 4 is seven point two five so then what you want to do is multiply 7 by 4 which gives you 28 that's the first number and then the decimal portion of the number point two five multiply that by 4 you should get one which is the other number what about I raised to the sixty-two this is I to the sixteen times I squared and I to the sixty is basically I to the fourth raised to the fifteen power I to the fourth is 1 I squared is negative 1 1 raised to anything is 1 so 1 times negative 1 is negative 1 let's try one more problem I raised to the 201 so let's work on this one so what two numbers should we use so let's use that process that we were dealing with earlier 201 divided by 4 this is equal to a 50 point 2 5 now if you multiply 4 by 50 this will give you 204 times 0.25 is equal to 1 so this is going to be 201 now 200 divided by 4 is 50 so you want to write it as I to the fourth raised to the 50th power times I to the first power and I to the fourth is 1 and 1 raised to the 50th is simply 1 so it's 1 times I which means the final answer is I so for these type of problems there's only four possible answers it's either I or I squared which is negative 1 I cube which is negative I or I to the fourth which is 1 so it's always going to simplify to one of these 4 values our next topic is adding and subtracting complex numbers so let's say if you have a problem that looks like this five plus 2i plus three plus 7i what would you do so in a situation like this all you need to do is combine like terms the real numbers are five and three so we can add those two numbers 5 plus 3 is 8 the imaginary number is 2i and 7i if we add them that is going to give us a 9 and that's all you need to do for this particular problem so let's try another example go ahead and add 7 plus 3i plus 6 plus 5i so 7 plus 6 is 13 3i plus 5i is a tie try this one four plus eight minus three minus five iron now we only have a 1 in front of the first parenthesis so if we multiply everything by one it's going to stay as four plus eight iron now here we need to distribute the negative sign so it's going to be negative three plus 5i so be careful when you have a negative because some people may forget to change this sign so now let's add the real numbers 4 minus 3 is 1 8i plus 5i is 39 so this is the answer 1 plus 39 now what is the absolute value of one plus 39 so don't forget the absolute value is going to be the square root of a squared plus B squared or 1 squared plus 13 squared 13 squared is 169 and if you add 1 to that you're going to get the square root of 170 now once 70 is divisible by 10 this is the square root of 17 and the square root of 10 there's no perfect square that goes into 170 so this is the final answer go ahead and simplify this problem 7 multiplied by 4 plus 3i minus 5 times 2 minus 6i so first so let's distribute 7 to 4 plus 3i so 7 times 4 is 28 and 7 times 3i is 21 now let's distribute the negative 5 negative 5 times 2 is negative 10 and negative 5 times negative 6 is positive 30 I so now let's combine like terms 28 minus 10 is 18 and 21 I plus 30 I is 51 I so this is the answer in standard form 1 a plus bi form you you try this four plus the square root of a negative 25 plus three minus the square root of negative 81 go ahead and simplify this expression so this is four plus we know the square root of 25 is 5 so the square root of negative 25 is 5 I and the square root of negative 81 is 9 I so we can add 4 plus 3 which will give us 7 and 5i minus 9 I is negative 4 I so this is the answer try this one seven minus the square root of negative 9 minus negative 4 plus the square root of negative 36 so this is going to be seven minus the square root of negative nine is nine nine and then if we distribute the negative sign this is going to be plus four and then minus the square root of negative 36 6i so 7 plus 4 is 11 and negative 9i minus 6i is negative 59 so this is it what about this one what is 8i x 4i what's the answer how would you simplify it eight times four is thirty-two I times I is I squared and as you recall I squared is negative one so the final answer is negative thirty-two try this one what is 5i raised to the second power so five squared is 25 times I squared so this is going to be 25 times negative 1 which is negative 25 now what about this what Street I times 5i times negative 7i so 3 times negative 7 is negative 21 and I times I times I is basically I to the third power twenty-one times five is 105 and I to the third is negative I so the final answer is positive 105 times I simplify this expression what is 5i x 4 minus 2 I go ahead and work on this example so let's distribute 5i times 4 is 20 I + 5 I times negative 2i 5 times negative 2 is negative 10 I times I is I squared so we could simplify the I squared part I squared is negative 1 and negative 10 times negative 1 is positive 10 now we need to put it in standard form that is an a plus bi form so we need to reverse the two numbers therefore the final answer is 10 plus 20 in standard form try this one what is 5 minus 3i times 4 plus 7i in this example we need to foil so what's 5 times 4 5 times 4 is 20 and then 5 times 7 I is 35 I negative 3i times 4 is negative 12 on and negative 3i times 7 I is negative 21 I squared so we can combine 35 I and 12 are 35 minus 12 is 23 negative 21 I squared that's a negative 21 times negative 1 which is positive 21 plus 20 so that's 41 so the final answer in standard form is 41 plus 23 I go ahead and work on this example multiply six minus 5i times three plus 8i so go ahead and for you 6 times 3 is 18 and 6 times 8 is 48 I negative 5i times 3 is negative 15 I and finally negative 5i times 8i is negative 40 I squared so let's combine 48 I and 59 so 48 minus 15 and that is equal to 33 or in this case 33 I and I squared is negative 1 so this is going to be 18 negative 40 times negative 1 is plus 40 and 18 plus 40 is 58 so the final answer is 58 plus 33 I now what if you were to see a problem like this 4 plus 5i squared what would you do to simplify in standard form so what this means is that you have to 4 plus 5i terms multiplied to each other so you want to write it out like this expand it and then 4 so 4 times 4 is equal to 16 and 4 times 5 on its 20 I 5i times 4 is also 20 I and 5i times 5i is 25 I squared so let's combine the two middle terms so that's going to be 20 plus 20 is 40 and 25 I squared is negative 25 so what is 16 minus 25 16 minus 25 is negative 9 so the final answer is negative 9 plus 40 I here's one for you go ahead and simplify this expression in standard form five the minus 3i raised to the third power so we need to expand it or write it three times so let's foil to two of these our binomials at one time so let's start with those two what's 5 times 5 5 times 5 is 25 and then we have 5 times negative 3i which is negative 15 I and then negative 3i times 5 which is also negative 59 and finally negative 3i times negative 3i which is positive 9 I squared and we still have another one on the outside so let's simplify this expression negative 15 I and negative 59 adds up to negative 30 I we still have the 25 and 9 I squared is negative 9 so now we can combine 25 minus 9 which is 16 so we have 16 minus 30 I times 5 minus 3i so at this point we'll need to foil these two expressions so 16 times 5 is 80 16 times negative 3i is negative 48 I and negative 30 I times 5 is negative 150 I and finally negative 30 I times negative 3i is positive 90 I squared so leaving my math is correct hopefully I didn't miss anything so once again let's add the two middle terms what's negative 48 plus negative 150 these two they're going to add to negative 198 times I and 90 I squared is basically negative 90 80 plus negative 90 is negative 10 so the final answer is negative 10 minus 198 times I now you need to know what's going to happen when you take a complex number and multiply it by its conjugate so consider the complex number 3 plus 4i the conjugate of this number is 3 minus 4i it has the same a and B value the only difference is this is a positive B value and this is a negative B value when you multiply a number by its conjugate you're going to get two terms initially and ultimately it's going to simplify to a real number the imaginary numbers will cancel so here's the quick way to get the answer it's going to be 3 times 3 which is 9 and 4 times negative 4i which is negative 16 I squared and I squared is negative 1 so that's plus 16 which is 25 that's the fast one but I'm going to show to you by foiling the entire problem so 3 times 3 is 9 3 times a negative 4i is negative 12i 4 times 3 is positive 12 are and finally 4 I times negative 4 is negative 16 I squared so notice that the middle terms cancel anytime you multiply a complex number by its conjugate and I squared is negative 1 so negative 16 I squared is negative 16 times negative 1 which is positive 16 and so the final answer is 25 so notice that there's no more imaginary numbers we just get a real number so let's try this example 5-2 I multiplied by its conjugate 5 plus 2 I so for this particular example we just need to multiply the first two terms 5 times 5 which is 25 and the last two terms negative 2i times 2i which is negative 4i squared so this is negative 4 times negative 1 which is plus 4 and 25 plus 4 is 29 now let's say if you were to see a question like this what is 3 times 7 I Square compared to 3 plus 7 I squared so what's the difference between the two and how would affect the way you would solve it so for the one on top notice that the 3 is multiplied by the 7 I so therefore this is equivalent to 3 squared times 7 I squared you can distribute the X 1 below you can't do that you can't say this is 3 squared plus 7 I squared doesn't work that way for the one on the bottom you need to foil it you need to expand it first as 3 plus 7i times 3 plus 7i and then foil so if you have an addition or a subtraction sign between the three and a seven time you have to foil it if you have a multiplication or division sign you can distribute the 2 you don't have to foil it make sure you understand the difference so for the example above 3 squared is 9 7 squared is 49 and I squared is just I squared so 9 times 49 is 441 times I squared which is negative 441 now for the example below we need to foil 3 times 3 is 9 3 times 7 is 63 I plus I messed up there 3 times 7 is not 63 3 times 7 is 21 and we're going to get another 21 I and finally 7 9 times 7 I is 49 I squared so we have 9 plus 42 I - 49 9 - 49 is negative 40 now what about dividing complex numbers let's say if we have four plus three are divided by five minus two are what would you do to simplify this expression if you're dividing complex numbers focus on the denominator which is five minus two I multiply the top and the bottom of the fraction by the conjugate of the denominator so if you see a minus sign make sure this is plus whatever you do to the bottom you must also do to the top in order that the fraction maintain its value so let's for them four times five is 20 4 times 2i is a tie 3i times five that's 15 I 3i times 2 I is plus 6i squared now because these two are conjugates we only need to multiply the first and the last term the two middle terms will cancel five times five is 25 and negative 2i times 2i is negative 4i squared so let's combine 8i and 59 8 plus 15 is 23 and 6i squared is negative 6 negative 4i squared it's going to be plus 4 so 20 minus 6 is 14 and 25 plus 4 is 29 now we need to put this in standard form so let's separate this fraction into two smaller fractions let's divide both numbers by 29 so this is 14 over 29 plus 23 over 29 times I so it's now an A plus bi form or standard form try this one divide 8 by 6 plus I so just like the last example we're going to multiply the top and the bottom by the conjugate of the denominator so let's distribute 8 to 6 are 8 times 6 is 48 and 8 times negative I is simply negative 8i on the bottom we need to foil but since they're conjugates we could just multiply the first 2 6 times 6 is 36 and the last two are times negative I which is negative I squared I squared is negative 1 so negative I squared must be positive 1 so this is equal to 48 minus 8i divided by 37 and now let's separate it into two smaller fractions so the final answer is 48 divided by 37 minus 8 over 37 times I so this is the answer in standard form so for the sake of practice try this one 7 + 2 I divided by 3 - on pause the video and work on this example so let's multiply by the conjugate of the denominator so on topless foil 7 times 3 is 21 7 times I that's 7 I 2 I times 3 6 i + 2 I times I which is 2 I squared on the bottom the first 2 3 times 3 is 9 and the last 2 negative I times I is negative I squared so let's add 7 i + 6 i that's going to be 13 i - y squared is negative 2 negative I squared is plus 121 minus 2 is 19 9 plus 1 is 10 so the final answer is 19 divided by 10 plus 13 divided by 10 times I what if you have a complex number and you wish to divide it by and imagine number what would you do in this problem so to simplify this expression your goal is to get rid of the imaginary number to do that multiply the top and the bottom by I if you can turn it into I squared or I to the fourth power you can get rid of the imaginary number so on top let's distribute our five times I is five I and negative 2i times I is negative 2 I squared on the bottom I times I is simply I squared so we have 5i negative 2 I squared is positive 2 I squared is negative 1 so this is 5 I divided by negative 1 which is negative 5i + 2 divided by negative 1 is negative 2 so this is equal to negative 2 minus 5 I so let's work on some more examples let's try 3 plus 2i divided by 7 9 so for this example we need to multiply the top and the bottom by I so on top if we distribute I it's going to be 3i plus 2i squared and on the bottom we're going to have 7i squared so 2 I squared is negative 2 7 I squared is negative 7 so I'm going to rewrite it as negative 2 plus 3i so it's going to be in standard form so negative 2 divided by 7 or negative 7 that's positive 2 over 7 and then 3i divided by negative 7 is negative 3 over 7 times I so this is the answer in standard form go ahead and simplify the expression so this one it looks weird but it's not that bad we just got to multiply the top and the bottom by I so on top is going to be nine I on the bottom I to the fourth and remember I to the fourth is 1 so the final answer is simply 9 I now what if you were to see a problem that looks like this 5 minus 2 I divided by 2 plus 3i squared what would you do to simplify so before we can multiply by the conjugate or by I we need to simplify this expression we need to put it in a plus bi form so let's foil 2 plus 3i times another 2 plus 3i so 2 times 2 is 4 2 times 3i is six I and 3i times 2 is also 6 I and then 3i times 3 is 9 I squared on the top everything is going to be the same 6i plus 6i is 12 I and 9 I squared is negative 9 so now we can combine 4 and negative 9 which is negative 5 so it's negative 5 plus 12 I on the bottom so now at this point we can multiply the top and the bottom by the conjugate of the denominator that is negative 5 minus 12r so on top let's distribute 5 times negative 5 is negative 25 5 times negative 12 on that's negative 60 I and negative 2i times negative 5 is positive 9 and finally negative 2i times negative 12i is positive 24 I squared so hopefully I didn't miss any negative signs or anything like that it's very easy to make a mistake things happen but I'm just double-checking my work so everything looks good so far on the bottom since they're conjugates of each other we could just multiply the first two and the last 2 negative 5 times negative 5 is positive 25 12v times negative 12i is negative 144 i squared so now let's combine negative sixty and ten nine negative sixty plus 10 is negative fifty twenty-four I squared is negative twenty-four and negative 144 I squared is plus 144 so now let's add negative twenty-five and negative 24 which is negative 49 25 plus 144 is 169 so the final answer for this problem is negative 49 divided by 169 minus 50 over 169 times on so as you can see whatever expression you have that's a complex number you can always put it in standard form there's always some technique that you can employ to put it in a plus bi form so let's say if you have two equations x squared minus 36 is equal to zero and also x squared plus 36 is equal to zero what would you do to solve for X in the first example we can factor it using the difference of squares method the square root of x squared is X the square root of 36 is 6 one will be positive and the other will be negative so therefore X is equal to negative 6 and positive 6 so that's what you could do if you have the difference of perfect squares if you have the sum of perfect squares it's going to be x plus 6i times X minus 6i so therefore X is equal to plus or minus 6 I notice that these two are conjugates of each other so you can check your answer by foiling so x times X is x squared and six I times negative six I is negative 36 I squared which is positive 36 so whenever you factor in the sum of perfect squares you're going to get imaginary numbers if it's a difference of perfect squares you're going to get real numbers so let's try some more examples solve for X so let's try this one 3x squared plus 48 is equal to 0 let's put it in factored form and then we'll solve for X so we can take out the GCF which is 3 3 x squared divided by 3 is x squared 48 divided by 3 is 16 so notice that we have the sum of perfect squares x squared is a perfect square 16 of them is a perfect square because you can take the square root of it the square root of x squared is X the square root of 16 is 4 but we're going to have 4 I and 4i 1 is going to be positive the other will be negative so therefore X is equal to positive 4 I and negative 4 I so let's try another one 4 x squared plus 100 is equal to 0 go ahead and work on that example so let's take out the greatest common factor 4x squared divided by 4 is x squared 100 divided by 4 is 25 so the factor it's going to be X plus 5 since the square root of 25 is 5 MX minus 5 on so to solve for X let's set each factor equal to 0 so in this example we need to subtract both sides by 5 I hear we need to add by 5 I so we can see that X is equal to negative 5i and positive 5i consider this one negative 9x squared minus 100 is equal to zero what would you do to solve this particular example what I would recommend is taken out the GCF which is negative 1 negative 9x squared divided by negative 1 is 9x squared negative 100 divided by negative 1 is plus 100 so notice that we have the sum of perfect squares 9 and 100 are perfect squares the square root of 9x squared is 3x the square root of 100 is 10 but it's going to be 10 9 since we know we have the sum of perfect squares so 1 is going to be positive and the other is going to be negative so we could write three equations let's set I mean two equations let's set each factor equal to zero so 3x plus 10 I is equal to zero and 3x minus 10 I is equal to zero so therefore X it's going to want to be negative ten I divided by three and it's also going to be positive 10 over 3 times on let's try this one x squared plus one over nine is equal to zero so here we have a fraction go ahead and solve for X the square root of x squared is X the square root of one over nine is 1/3 now let's not forget to put the imaginary numbers so we're going to have a positive and a negative sign so we could see that X is equal to negative 1 over 3 times I and positive 1 over 3 times R and that's all you got to do for that particular example so how would you solve for this particular equation 3x squared plus 4x plus 7 now typically you would try to see if you can factor this expression since we have a trinomial 3 times 7 is 21 if we could find two numbers that multiply to 21 but that add to the middle term 4 then this expression is factorable the only factors of 21 are 1 in 21 and 3 and 7 which none of these add up to 4 we also have negative 1 and negative 21 and negative 3 and negative 7 now that doesn't add to 21 unless we have like negative 3 and positive 7 that would add to 4 but negative 3 times 7 doesn't multiply it's a positive 21 it multiplies to negative 21 so this expression is not factorable so therefore the only way to solve it is either to complete the square or to use the quadratic formula this equation is in standard form ax squared plus BX plus C so a is 3 B is 4 cs7 so let's use the quadratic formula X is equal to negative b plus or minus square root b squared minus 4ac the by 2a so B is 4 which means that B squared 4 squared is 16 a is 3 C is 7 divided by 2 times a or 2 times 3 so this is going to be negative 4 plus or minus 16 now 3 times 7 is 21 and 21 times 4 is 84 divided by 6 so now what is 16 minus 84 16 minus 84 is negative 68 can we simplify the square root of 68 it turns out that we can the square root of 68 is basically 4 the square root of 4 times the square root of 17 4 times 17 is 68 now let's not forget the negative 1 the square root of 4 is 2 and the square root of negative 1 is I so this is what we have at this point so that's two answers but let's separate it into two fractions so it's negative 4 divided by 6 plus or minus 2 radical 17 divided by 6 times I so we can reduce 4 over 6 if we divide both numbers by 2 it's going to be negative 2 over 3 and 2 over 6 we can reduce them 6 divided by 2 is 3 but it's going to be on the bottom so it's the square root of 17 divided by 3 times I so the two answers are negative 2 over 3 plus root 17 over 3 times I that's in a plus bi form and the other answer is negative 2 over 3 minus root 17 over 3 times.i let's try another example try this one 2x squared minus 3x plus 9 go ahead and solve for X so let's use the quadratic equation one more time it's a negative b plus or minus square root b squared minus 4ac divided by 2a so in this problem B is negative three so B squared or negative three squared that's going to be 9 a is 2 and C is 9 divided by 2a or 2 times 2 so negative times negative 3 is positive 3 4 times 2 is 8 8 times 7 I mean 8 times 9 is 72 and 2 times 2 is 4 so what is 9 minus 72 9 minus 72 is negative 63 so notice that we could simplify route 63 route 63 63 is basically 9 times 7 and let's not forget the square root of negative 1 so the square root of 9 is 3 and the square root of negative 1 is I so this is what we now have so let's separate it into two fractions so this is going to be 3 over 4 plus or minus 3 radical 7/4 times I and so the two answers are 3 over 4 plus 3 root 7 over 4 times I and the second answer it's 3 over 4 minus 3 root 7 divided by 4 times I so these are the two imaginary solutions now what if you were to see an equation that looks like this 2x times X minus 3 is equal to 5 or rather let's say negative 5 what would you do to solve for X so notice that this is a quadratic equation but not in standard form so we got to put it in standard form before you can use the quadratic formula so 2x times X is 2x squared 2x times negative 3 is negative 6x and let's add 5 to both sides so it's 2x squared minus 6x plus 5 which is equal to 0 so now we can use the quadratic formula now that it's in standard form so now B is negative 6 and then we have B squared we're negative 6 squared minus 4 and a is 2 C is 5 divided by 2a or 2 times 2 so this is going to be 6 plus or minus square root negative 6 squared is 36 2 times 5 is 10 and 10 times negative 4 is negative 40 and 2 times 2 is 4 so let's make some space so 36 minus 40 is negative 4 and the square root of negative 4 the square root of 4 is 2 to the square root of negative 4 is 2i so let's separate it into two fractions so this is going to be 6 over 4 plus or minus 2 over 4 times our 6 over 4 reduces to 3 over 2 if you divide the top and bottom by 2 and 2 over 4 reduces to 1 over 2 times R so therefore we have two answers 3 over 2 plus 1/2 I and the second answer 3 over 2 minus 1/2 I and that's it for this problem now what if you were to see an equation like this 6x plus 2i is equal to 18 plus 12 yr so there's two variables x and y what can you do to solve for x1 now you need to keep in mind that these complex numbers have two components the real component and the imaginary component the real component on the left side is 6x because it doesn't have an eye attached to it the real component on the right side is 18 so therefore we could say that 6 X must be equal to 18 now the imaginary component contains an eye the imaginary component on the left side is 2i and on the right side is 12 Y I so therefore we could say that 2 on is equal to 12 yr on the left we could divide by 6 so we could say X is equal to 3 and on the right side to solve for y we could divide both sides by 12 on to get Y by itself so on the right 12 I will cancel leaving us with Y on the left the I variable will cancel and it's 2 over 12 if you divide it backwards 12 divided by 2 is 6 so 2 over 12 must be 1 over 6 so this is the answer X is equal to 3 and Y is equal to 1 over 6 try this one 3x plus 4y is equal to 15 plus 16 yr so let's set 3x equal to 15 since both of those terms are real numbers and let's set the imaginary parts equal to each other so for I is equal to 16 Y I so let's divide by 3 15 divided by 3 is 5 so X is 5 and for the second equation let's divide by 16 I so on the right side all we have is just Y on the left side the I variables cancel 16 divided by 4 is 4 so 4 of the 16 must be 1 over 4 and so those are the answers try this one what is X plus 4 I multiplied by X minus 4 and let's say that's equal to 41 what is the value of X so we have two conjugates multiplied to each other so we can multiply the first and last term's X times X is x squared and 4 times negative 4 is negative 16 I squared and since I squared is negative 1 negative 16 I squared is plus 16 so now what we need to do is subtract both sides by 16 41 minus 16 is 25 so we could take the square root of 25 therefore X is going to be plus or minus 5 so that's the answer to the problem now let's say if you're given the imaginary solutions to an equation and you're asked to write that equation what would you do so first you want to write it in factored form if your answer is 3i you want to write X minus 3i if it's negative 3i write X plus 3i of course this is probably equal to 0 you may or may not need that 0 and then you want to foil since these are conjugates of each other we could just multiply the first and the last two terms so it's going to be x squared times negative 9i squared which is x squared plus 9 so that's how you can find the equation if you're given the solutions so let's try another example so let's say if you're given just one solution for I what's the equation now for quadratic equation imaginary numbers they come in pairs so if you have 4i you must also have the conjugate negative 4i so therefore we can write X minus 4i times X plus 4i and we know it's going to be x squared minus 16 I squared which is x squared plus 16 so basically you're working backwards now let's say if you're given one of two imaginary solutions let's say 4 plus 3i and you want to write the equation first need to find the other imaginary solution which is going to be the conjugate for minus 3i so how can we write these two numbers in factored form so here's what you need to do notice that you have a plus 4 change the sign of four and it's going to be minus four and then change the sign of plus 3i it's going to be monastry on do the same thing for the other side so we have a positive 4 change it to negative 4 negative 3 I change it to positive 3i and so now we need to foil so there's an easy way and there's a long way we're going to do it the long way first and then we'll do it the easy way so you'll see why it works here we have three terms three terms initially when we foil it we should get nine terms and then we'll simplify so x times X is x squared X times negative four negative four X x times 3i that's going to be three X I and then negative 4 times X negative four X negative 4 times negative 4 is 16 and negative 4 times 3i is negative 12 oh and then we have negative 3i times X which is negative 3x I and then negative 3 times negative four that's positive 12 R and finally negative 3i times 3i is negative 9i squared so now what terms cancel we can cancel the three X I and the negative three X on we can also cancel the negative 12 I and a 12 R so what do we have left over so we have x squared and these two combine to form negative 8x plus 16 minus 9i squared negative 9i squared is positive 9 and 16 plus 9 is 25 so this is the quadratic equation x squared minus 8x plus 25 now let's start back from our original factored expression which was X minus 4 plus 3i and X plus 4 I mean minus 4 but minus 3i so our goal is to get this answer x squared minus 8x plus 25 so another way in which you can do it is you're going to multiply X minus 4 times X minus 4 which is X minus 4 squared and then you're going to multiply these two 3i times negative 3i so when you foil X minus 4 squared because it's a perfect square it's going to be x squared and it's going to be negative 4x plus negative 4x and negative 4 squared is going to be 16 negative 4 times negative 4 and 3 R times negative 3i is negative 9i squared so this is going to be x squared minus 8x plus 16 plus 9 which turns into this so that's how you can do it the fast way let's try another example so given these two solutions go ahead and write the quadratic equation so let's write it in factored form since we have positive 5 it's going to be X minus 5 plus 2i and X minus 5 minus 2i so this is the same as X minus 5 squared plus 2i times negative 2i so X minus 5 squared that's going to be X minus 5 times X minus 5 which is x squared minus 5x minus 5x and negative 5 times negative 5 is 25 and 2i times negative 2i is negative 4i squared so this is going to be negative 10x plus 25 negative 4i squared is +4 so the final answer is x squared minus 10x plus 29 so this is it now what if you were to see something like this X is equal to the square root of 7 plus 3 R what's the other answer the other answer is going to be root 7 but minus 3 R so let's go ahead and convert this into a quadratic equation using both techniques so this is going to be since we have a positive in front of the radical 7 it's going to be X in minus root 7 minus 3i and X minus root 7 but plus 3 R so let's foil 8 the long way x times X is x squared X times negative root 7 that's negative root 7 X and + 3 X I and then negative root 7 times X and then this is going to be negative reach 7 times the negative root 7 which is simply negative 7 because that's that's actually positive 7 the two negative signs cancel root 7 times root 7 is the square root of 49 which is 7 and negative root 7 times 3i that's going to be negative 3 root 7i negative 3i times X is negative 3x I negative 3i times negative 7 negative 3 well it's going to be positive now positive 3 root 7 times I'm you got to be careful it's very easy to like make a mistake negative 3i times 3i is negative 9i squared so now let's cancel the terms that can be cancelled that's 3x I and also our 3 root 7i so we're left with x squared these two we could combine that's going to be 1 radical 7 plus 1 radical 7 is 2 radical 7 so this is going to be negative 2 radical 7 times X and then +7 negative 9i squared is going to be plus 9 so the final answer is x squared minus 2 root 7 times X plus 16 now let's get the same answer using the other technique so in factored form it was X minus root 7 minus 3i times X minus root 7 plus 3i so this is equivalent to being an X minus root 7 squared and then plus 3i times negative 3i so X minus root 7 times X minus root 7 that's going to be x squared minus root 7 X minus another root 7 X and negative root 7 times negative root 7 is positive 7 and 3i times negative 3i is negative 9i squared so we can see that these two they're going to add to negative 2 root 7 X and then this is going to be 7 negative 9i squared is plus 9 which these two will become 16 so you get the same answer now let's say if you're given a quadratic equation in standard form ax squared plus BX plus C and you want to find the sum and the product of the roots in a quadratic equation there's usually two roots two real solutions or two imaginary solutions or one real solution that's the same but two answers how can we find the sum and the product of the roots if we're given the quadratic equation the equation is this the sum is equal to negative B divided by a and the product is seen divided by 8 so let's try an example let's say if we have the quadratic equation x squared plus 36 equals 0 what is the sum and the product of the roots in this equation so this is 1 x squared plus is 0 x plus 36 which is equivalent to x squared plus 36 so we can see that beam is 0 so the sum is going to be negative 0 divided by a which a is 1 so the sum is 0 the product is going to be C over a C is 36 so it's 36 over 1 which is 36 now let's prove it so the equation x squared plus 36 can be factored as X plus 6 I times X minus 6 are therefore the roots are negative 6 I and positive 6 I so if we are looking for the sum of the roots the sum of the roots is just adding the two answers negative 6i plus 6i adds up to 0 now the product we just got to multiply negative 6i x 6i will give us the product which is negative 36 I squared and since I squared is negative 1 this is going to be positive 36 let's try another example x squared minus 4x plus 29 so given this quadratic equation with three terms in other words a trinomial find the sum and the product of the roots so the sum of the roots is negative B divided by a the product is simply C divided by a so use the equation and also find the solutions and confirm the answer so let's use the equation first B is negative 4 a is 1 so the sum is 4 which I'm going to write it right here the product is C over a where C is 29 a is 1 so the product is 29 so now let's find the solutions and let's confirm that answer so let's solve for X using the quadratic equation negative b plus or minus square root b squared minus 4ac divided by 2a so B is negative 4 and B squared negative 4 squared negative 4 times negative 4 is 16 a is 1 C is 29 divided by 2a or 2 times 1 negative times negative 4 is positive 4 four times twenty nine and that's going to be one 16/2 16 minus 116 is negative 100 now the square root of 100 we know it's 10 so the square root of negative 100 must be 10 times on which is the nice number so now let's separate into two fractions that's going to be 4 over 2 plus a minus 10 over 2 and 4 divided by 2 is 2 10 over 2 is 5 so there's two answers 2 plus 5i and 2 minus 5i so these are the two complex solutions now that we have the solutions we can confirm if the sum is 4 and if the product is 29 so let's go ahead and do that to find a sum we need to add the two solutions 2 plus 5i plus 2 minus 5r so 2 plus 2 is 4 5i plus negative 5i they cancel at 0 so the sum is indeed for now let's find the product of the two solutions so the product is going to be 2 plus 5i x 2 minus 5r now because these are conjugates of each other we can multiply the first two and the last two 2 times 2 is 4 5 r times negative 5i is negative 25 I squared negative 25 I squared is plus 25 since I squared is negative 1 and 4 plus 25 is 29 you