Answer
The quadratic equation: ${{x}^{2}}-9x+22$
Work Step by Step
The quadratic equation is as shown below:
$y=a{{x}^{2}}+bx+c$
The points are $\left( 2,8 \right),\ \left( 4,2 \right),\ and\ \left( 6,4 \right).$ The task here is to find the equations in variables a, b and c and solve for the values of a, b, and c.
Substitute $x=2$ and $y=8:$
$\begin{align}
& 8=a{{\left( 2 \right)}^{2}}+b\left( 2 \right)+c \\
& 8=4a+2b+c
\end{align}$
Substitute $x=4$ and $y=2:$
$\begin{align}
& 2=a{{\left( 4 \right)}^{2}}+b\left( 4 \right)+c \\
& 2=16a+4b+c
\end{align}$
Substitute $x=6$ and $y=4:$
$\begin{align}
& 4=a{{\left( 6 \right)}^{2}}+b\left( 6 \right)+c \\
& 4=36a+6b+c
\end{align}$
The system of equations in variables a, b and c is as shown below:
$4a+2b+c=8$
$16a+4b+c=2$
$36a+6b+c=4$
Mark these equations as shown below:
$4a+2b+c=8$, Mark it as equation (I)
$16a+4b+c=2$, Mark it as equation (II)
$36a+6b+c=4$, Mark it as equation (III)
By multiplying equation (I) by $-4$ and eliminating a from equations (I) and (III), we get
$\begin{align}
& \text{ }16a+4b+\text{ }c=\text{ }2 \\
& \underline{-16a-8b-4c=-3\text{2}} \\
& \text{ }-\text{4}b-3c=-30 \\
\end{align}$
Mark this equation as
$-\text{4}b-3c=-30$ (IV)
By multiplying equation (III) by −9 and eliminating a from equations (II) and (III), we get
$\begin{align}
& \text{ }36a+\text{ }6b+\text{ }c=\text{ }4 \\
& \underline{-36a-18b-9c=-\text{72}} \\
& \text{ }-12b-8c=-68 \\
\end{align}$
Mark this equation as
$-12b-8c=-68$ (V)
By multiplying equation (IV) by −3 and then adding equations (IV) and (V), we get the value of c:
$\begin{align}
& -12b-8c=-68 \\
& \underline{-12b+9c=\text{ }90} \\
& \text{ }c=\text{ }22 \\
\end{align}$
Now, putting the value of c in equation (IV) to get the value of b:
$\begin{align}
& -4b-3\left( 22 \right)=-30 \\
& -4b-66=-30
\end{align}$
Now, adding 66 to both sides:
$\begin{align}
& -4b-66+66=-30+66 \\
& -4b=36
\end{align}$
By dividing both sides by −4 we get:
$\begin{align}
& \frac{-4b}{-4}=\frac{36}{-4} \\
& \text{ }b=-9
\end{align}$
Substitute the values of b and c in equation (I) to get the value of a:
$\begin{align}
& 4a+2\left( -9 \right)+22=8 \\
& 4a-18+22=8 \\
& 4a+4=8
\end{align}$
Now, subtract $4$ from both sides to get:
$\begin{align}
& 4a+4-4=8-4 \\
& 4a=4
\end{align}$
By dividing both sides by 4, we get:
$\begin{align}
& \frac{4a}{4}=\frac{4}{4} \\
& \text{ }a=1
\end{align}$
Hence, the quadratic equation is ${{x}^{2}}-9x+22$.
