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Without solving the equation negative three π₯ squared minus 16π₯ plus 63 equals zero, find the sum of its roots.
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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 what weβre told to do is find the sum of its roots.
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Well, instinct might think, well, letβs solve this and then weβll be able to find each root and add them together.
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However, the question states it does not want us to solve the equation.
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So how else can we do this?
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Well, what we actually have are a couple of formulae that can help us.
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And they can help us by using what we have for our coefficients of our π₯ terms and also the numerical term in our quadratic.
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First of all, we know that the sum of roots is equal to negative π over π.
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We also know the product of the roots is equal to π over π.
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We donβt need this for this question, but it was worth having to look at.
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So what weβre gonna use is the first one, the sum of the roots is equal to negative π over π.
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Well, we can see from our quadratic that our π, π, and π are negative three, negative 16, and 63, respectively.
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And what we need to do is remember the signs which weβve done here.
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So weβve got two negatives and a positive.
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Now, we only need π and π to find out the sum of the roots.
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So letβs plug these in.
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Well, when we plug them in, what weβre gonna get is the sum of the roots is equal to negative negative 16 over negative three.
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Well, negative negative 16 is gonna give us positive 16.
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Well, positive 16 over negative three gives us negative 16 over three.
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So we can say this is the sum of the roots of the equation negative three π₯ squared minus 16π₯ plus 63 equals zero.