Answer
The required quadratic equation is $ y=3{{x}^{2}}-4x+5$.
Work Step by Step
We know that the graph is passing through the point $\left( 1,4 \right);$ the equation becomes:
$ a+b+c=4$
Now, the graph is passing through the point $\left( 3,20 \right);$ the equation becomes:
$9a+3b+c=20$
Then the graph is passing through the point $\left( -2,25 \right);$ the equation becomes:
$4a-2b+c=25$
Subtract equation $ a+b+c=4$ from equation $9a+3b+c=20$, given that
$\begin{align}
& 8a+2b=16 \\
& b=8-4a
\end{align}$
Subtract equation $ a+b+c=4$ from equation $4a-2b+c=25$ to get:
$\begin{align}
& 3a-3b=21 \\
& a-b=7
\end{align}$
Substitute the value of b from equation $ b=8-4a $ in equation $ a-b=7$ as given below:
$\begin{align}
& a-\left( 8-4a \right)=7 \\
& a-8+4a=7 \\
& 5a=15 \\
& a=3
\end{align}$
From equation $ b=8-4a $,
$\begin{align}
& b=8-4\times 3 \\
& =-4
\end{align}$
From equation $9a+3b+c=20$,
$\begin{align}
& c=4-a-b \\
& =4-3+4 \\
& =5
\end{align}$
Therefore, $ a=3,b=-4,\text{ and }c=5$; put these values in the quadratic equation
$ y=a{{x}^{2}}+bx+c $ as given below:
$ y=3{{x}^{2}}-4x+5$
Hence, the quadratic equation is $ y=3{{x}^{2}}-4x+5$.
