Answer
The resultant quadratic equation is ${{x}^{2}}-6x+8$.
Work Step by Step
The given quadratic equation is as shown below:
$y=a{{x}^{2}}+bx+c$
The given points are $\left( 1,3 \right),\left( 3,-1 \right),\left( 4,0 \right)$. In order to find the quadratic equation, find the values of a, b, and c.
Putting in the value of $x=1$ and $y=3$: we get,
$\begin{align}
& 3=a{{\left( 1 \right)}^{2}}+b\left( 1 \right)+c \\
& 3=a+b+c
\end{align}$
Putting in the value of $x=3$ and $y=-1$: we get,
$\begin{align}
& -1=a{{\left( 3 \right)}^{2}}+b\left( 3 \right)+c \\
& -1=9a+3b+c
\end{align}$
Putting the value of $x=4$ and $y=0$: we get,
$\begin{align}
& 0=a{{\left( 4 \right)}^{2}}+b\left( 4 \right)+c \\
& 0=16a+4b+c
\end{align}$
The resultant system of equations having variable a, b and c is as shown below:
$\begin{align}
& a+b+c=3 \\
& 9a+3b+c=-1 \\
& 16a+4b+c=0
\end{align}$
By multiplying equation (I) with $-9$ and eliminate a equation (I) and (II): we get,
$\begin{align}
& 9a+3b+\text{ }c=-1 \\
& -9a-9b-9c=-27
\end{align}$
$-6b-8c=-28$ (IV)
Multiply equation (I) with -16 and eliminate a equation (I) and (III): we get,
$\begin{align}
& 16a+\text{ }4b+\text{ }c=0 \\
& -16a-16b-16c=-48
\end{align}$
$-12b-15c=-48$ (V)
Now, multiply the equation (IV) by 2 and then add to get the value of c,
$\begin{align}
& -12b-15c=-48 \\
& 12b+16c=56
\end{align}$
$c=8$
Putting in the value of c in equation (IV), we get,
$\begin{align}
& -6b-8\left( 8 \right)=-28 \\
& -6b-64=-28
\end{align}$
By adding 64 to both sides: we get,
$\begin{align}
& -6b-64+64=-28+64 \\
& -6b=36
\end{align}$
Now, divide both sides by $-6$ to get the value of b,
$\begin{align}
& \frac{-6b}{-6}=\frac{36}{-6} \\
& \text{ }b=-6 \\
\end{align}$
Putting the values of b and c in equation (I) to get the value of a,
$\begin{align}
& a+\left( -6 \right)+8=3 \\
& a+2=3
\end{align}$
Subtract 2 from both sides to get,
$\begin{align}
& a+2-2=3-2 \\
& \text{ }a=1 \\
\end{align}$
Now substitute the values of a, b and c to get the equation ${{x}^{2}}-6x+8$.
Hence, the quadratic equation is ${{x}^{2}}-6x+8$.
