Question

Write the equation of a quadratic

function a single root at at x = 5 and
goes through the point (-1,12).

Answer 1

if it only has one root and is a 2nd degree polynomial, that means that the only one root has multiplicity of 2, or namely is repeated twice.

`\begin{cases} x = 5 &\implies x -5=0\\ x = 5 &\implies x -5=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x -5 )( x -5 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that} \begin{cases} x=-1\\ y=12 \end{cases} \\\\\\ a ( -1 -5 )( -1 -5 ) = 12\implies a36=12\implies a=\cfrac{12}{36}\implies a=\cfrac{1}{3} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1}{3}(x-5)(x-5)=y\implies \cfrac{1}{3}(x^2-10x+25)=y\implies \boxed{\cfrac{x^2}{3}-\cfrac{10x}{3}+\cfrac{25}{3}=y}`