<s>[INST] Solve the equation $h(x) = f(x) + g(x) = 0$, $f(x)$ and $g(x)$ having roots that are negatives of each other. Let $f(x) = x^2 +bx+ 9$ and let $g(x) = x^2 +ax+c, a, b, c ∈ R$. The roots of $f(x) = 0$ and
$g(x) = 0$ are negatives of each other. If $h(x) = f(x)+g(x)$, then solve the equation $h(x) = 0$.
I'm not sure how to solve this at all, maybe Vieta's formulas for quadratics can help.
 [/INST] $f(x) = x^2 +bx+ 9;\;g(x) = x^2 +ax+c$
The sum of the roots of $f(x)$ is $s=-b$ and their product is $p=9$
The sum the roots of $g(x)$ is $-a$ and their product is $c$
So we must have $a=-b;\;c=9$
$f(x)=x^2+bx+9;\;g(x)=x^2-bx+9$
$h(x)=f(x)+g(x)=2x^2+18=0\to x=\pm 3i$
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