Answer
$x^2-50=0$
Work Step by Step
With solutions of $5\sqrt{2}$ and $-5\sqrt{2},$ the factored form of the quadratic equation with these solutions is
\begin{align*}
\left(x-5\sqrt{2}\right)\left(x-(-5\sqrt{2})\right)&=0
\\
\left(x-5\sqrt{2}\right)\left(x+5\sqrt{2}\right)&=0
.\end{align*}
Using $(a+b)(a-b)=a^2-b^2$ or the special product of the sum and difference of like terms, the equation above is equivalent to
\begin{align*}\require{cancel}
(x)^2-\left(5\sqrt{2}\right)^2&=0
\\
x^2-25(2)&=0
\\
x^2-50&=0
.\end{align*}
Hence, the quadratic equation whose solutions are $5\sqrt{2}$ and $-5\sqrt{2}$ is $x^2-50=0$.
