y6pbzhm62d y6pbzhm62d
  • 10-08-2022
  • Mathematics
contestada

Let $r$ and $s$ be the roots of $3x^2 + 4x - 12 = 0.$ Find $r^2 + s^2.$

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LammettHash
LammettHash LammettHash
  • 10-08-2022

We can factorize the quadratic as

[tex]3x^2 + 4x - 12 = 3 (x - r) (x - s)[/tex]

and expand the right side to get

[tex]3x^2 + 4x - 12 = 3x^2 - 3(r + s) x + 3rs[/tex]

[tex]\implies r + s = -\dfrac43 \text{ and } rs = -4[/tex]

Then we find

[tex](r + s)^2 = r^2 + 2rs + s^2 \implies r^2 + s^2 = \left(-\dfrac43\right)^2 - 2(-4) = \boxed{\dfrac{88}9}[/tex]

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