Answer
$2x^2-13x+20=0$
Work Step by Step
With the given roots, $
2.5\text{ and }4
,$ then
\begin{align*}
x&=2.5
\\\\&\text{ OR }\\\\
x&=4
.\end{align*}
Using the properties of equality, the equations above are equivalent to
\begin{align*}
x&=2.5
\\
x&=\dfrac{5}{2}
\\
2(x)&=\left(\dfrac{5}{2}\right)2
\\
2x&=5
\\
2x-5&=0
\\\\&\text{ OR }\\\\
x&=4
\\
x-4&=0
.\end{align*}
A quadratic equation that has the two given roots is
\begin{align*}
(2x-5)(x-4)&=0
.\end{align*}
Using $(a+b)(c+d)=ac+ad+bc+bd$ or the FOIL method, the equation above is equivalent to
\begin{align*}
2x(x)+2x(-4)-5(x)-5(-4)&=0
\\
2x^2-8x-5x+20&=0
\\
2x^2-13x+20&=0
.\end{align*}
Hence, a quadratic equation that has the given pair of numbers as roots is $
2x^2-13x+20=0
$.
