Answer
$4x^2-8x+3=0$
Work Step by Step
Given the quadratic equation $ax^2+bx+c=0,$ the sum and product of its roots are given by the following ratios:
\begin{align*}
\text{Sum: }&-\dfrac{b}{a}
\\\\
\text{Product: }&\dfrac{c}{a}
.\end{align*}
Given that the sum is $
2
$ and the product is $
\dfrac{3}{4}
,$ then
\begin{align*}
\text{Sum: }\\
-\dfrac{b}{a}&=2
\\&=-\dfrac{-8}{4}
\\\\
\text{Product: }\\
\dfrac{c}{a}&=\dfrac{3}{4}
.\end{align*}
Hence, $a=
4
,$ $b=
-8
,$ and $c=
3
.$ The quadratic equation that satisfies the given sum and product is
\begin{align*}
4x^2-8x+3=0
.\end{align*}
