WEBVTT
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Determine the solution set of π₯ squared minus eight π₯ plus 185 equals zero over the set of complex numbers.
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So what weβre gonna do here is solve this quadratic equation.
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And the way Iβm gonna approach that is Iβm gonna subtract 185 from each side of that equation first.
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And that leaves me with π₯ squared minus eight π₯ on the left-hand side and negative 185 on the right-hand side.
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What it leaves me with is one π₯ squared on the left-hand side minus eight π₯.
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Now, weβre gonna use our experience of completing the square.
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To have an initial guess so that can be expressed as π₯ minus four all squared or π₯ minus four times π₯ minus four.
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Because we had one π₯ squared, we just got one π₯ and one π₯ here.
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And because we had negative eight π₯ here, weβre gonna take half of negative eight, which is negative four.
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Now, when I multiply that out, I get π₯ times π₯ which is π₯ squared, π₯ times negative four which is negative four π₯, negative four times π₯ which is another negative four π₯, and negative four times negative four which is positive 16.
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So π₯ squared minus four π₯ minus another four π₯ plus 16.
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Well, negative four π₯ take away another four π₯ is negative eight π₯.
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So it turns out that I guess weβre slightly wrong.
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This here, π₯ minus four all squared, isnβt quite the same as π₯ squared minus eight π₯.
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Yes, it has got π₯ squared minus eight π₯ in it, but itβs also got an extra plus 16 β an extra add 16.
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So if I took that 16 away from π₯ minus four times π₯ minus four, then I would just be left with π₯ squared minus eight π₯.
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So π₯ minus four all squared minus 16 is the same as π₯ squared minus eight π₯ and thatβs equal to negative 185.
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Well, Iβm trying to solve this for π₯.
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Iβm gonna add 16 to both sides, and this gives me π₯ minus four all squared on the left-hand side and negative 169 on the right-hand side.
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And to solve this for π₯, we are going to have to take square roots of both sides so that when itβs here π₯ minus four all squared, weβve just got π₯ minus four.
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But we encounter a slight problem because there are no real solutions to the square root of negative 169.
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So Iβm gonna re-express negative 169 as 169 times negative one.
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So taking the square root of the left-hand side gives me just π₯ minus four and over on the right-hand side, the square root of 169 is 13.
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So really there are two solutions here.
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Negative 13 would also give us 169 if we squared it.
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And then we can use a definition of imaginary numbers; π squared is equal to negative one.
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So π is the square root of negative one, to say that the square root of negative one is π.
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So π₯ minus four is equal to positive or negative 13π.
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Now if I add four to each side of the equation, I get π₯ on the left-hand side and plus or minus 13π plus four or four plus or minus 13π on the right-hand side.
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Now if we look back at the question, it asked us to determine the solution set.
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So I need to write this out in solution set notation.
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The solution set contains two values four plus 13π and four minus 13π.