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Given that πΏ and πΏ squared are the roots of the equation four π₯ squared plus ππ₯ plus 32 equals zero, find the value of π.
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So what we have here is a quadratic equation in the form ππ₯ squared plus ππ₯ plus π equals zero.
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And we know that the roots of the equation or solutions are πΏ and πΏ squared.
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But how are we gonna use this and the equation weβve got to help us find out the value of π?
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Well, what we have are a couple of relationships to help us because we have some relationships that are to do with π, π, and π, our coefficient of π₯ squared, our coefficient of π₯, and our numerical term.
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So, first of all, we know that the sum of the roots is equal to negative π over π.
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And then we also know that the product of the roots is equal to π over π.
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So, first of all, what we can do is to identify our π, π, and π.
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Well, our π is four, our π is just π, and our π is 32.
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Well, first of all, what weβre gonna use is the product of the roots.
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And we know that the product of the roots is equal to π over π.
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Well, then what we can say is that the product of the roots, so thatβll be the two roots multiplied together β well, we know the roots are πΏ and πΏ squared, so itβs πΏ multiplied by πΏ squared β is going to be equal to our π, which is 32, over our π, which is four.
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Well, therefore, what weβve got is πΏ cubed is equal to eight.
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So now what we need to do is cube root both sides of the equation.
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Well, what this is gonna give us is πΏ is equal to two.
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And thatβs cause the cube root of πΏ cubed is πΏ and the cube root of eight is two.
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Okay, great.
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So we now know that one of the roots is two.
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So now what we need to do is work out the value of π.
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But how are we going to do this?
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Well, what weβre gonna do is move on to our relationship that we know about the sum of the roots.
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And that is that negative π over π is the sum of the roots.
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Well, therefore, what we can say is that πΏ plus πΏ squared, because thatβs gonna be the sum of our roots, is equal to negative π over four.
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Well, we know what the value of πΏ is cause we just worked it out.
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So weβre gonna substitute this in as well.
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So when we do this, what weβre gonna have is two plus two squared equals negative π over four.
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Well, two plus two squared is just six.
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So we got six equals negative π over four.
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So therefore, what we do is multiply both sides of the equation by four and we get 24 is equal to negative π.
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So then we divide through by negative one.
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And what we get is negative 24 is equal to π.
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So we arrive at our final answer.
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And what we can say is that the value of π is negative 24.