Answer
$2x^2+x-6=0$
Work Step by Step
With the given roots, $
\dfrac{3}{2}\text{ and }-2
,$ then
\begin{align*}
x&=\dfrac{3}{2}
\\\\&\text{ OR }\\\\
x&=-2
.\end{align*}
Using the properties of equality, the equations above are equivalent to
\begin{align*}
x&=\dfrac{3}{2}
\\
2(x)&=\left(\dfrac{3}{2}\right)2
\\
2x&=3
\\
2x-3&=0
\\\\&\text{ OR }\\\\
x&=-2
\\
x+2&=0
.\end{align*}
A quadratic equation that has the two given roots is
\begin{align*}
(2x-3)(x+2)&=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(2)-3(x)-3(2)&=0
\\
2x^2+4x-3x-6&=0
\\
2x^2+x-6&=0
.\end{align*}
Hence, a quadratic equation that has the given pair of numbers as roots is $
2x^2+x-6=0
$.
