Question

Write the quadratic function in factored form given the parabola has roots of 9 and -6 and passes through the point (-8,-102)

Answer 1

`\begin{cases} x = 9 &\implies x -9=0\\ x = -6 &\implies x +6=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -9 )( x +6 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that} \begin{cases} x=-8\\ y=-102 \end{cases}`

`a ( -8 -9 )( -8 +6 ) = -102\implies a(-17)(-2)=-102 \implies 34a=-102 \\\\\\ a=\cfrac{-102}{34}\implies \boxed{a=-3} \hspace{9em} \stackrel{\textit{plugging into our equation} }{-3(x-9)(x+6)=y} \\\\\\ -3(x^2-3x-54)=y\implies {\Large \begin{array}{llll} -3x^2+9x+162=y \end{array}}`