Answer
$y=3x^2-4x+5$
Work Step by Step
We have to determine $a,b,c$ so that the graph of the function $y=ax^2+bx+c$ passes through the points $(1,4),(3,20),(-2,25)$.
We can write the system:
$\begin{cases}
a(1)^2+b(1)+c=4\\
a(3)^2+b(3)+c=20\\
a(-2)^2+b(-2)+c=25
\end{cases}$
$\begin{cases}
a+b+c=4\\
9a+3b+c=20\\
4a-2b+c=25
\end{cases}$
We will use the addition method. Multiply Equation 1 by -1 and add it to Equation 2, then to Equation 3 to eliminate $c$:
$\begin{cases}
9a+3b+c-(a+b+c)=20-4\\
4a-2b+c-(a+b+c)=25-4
\end{cases}$
$\begin{cases}
9a+3b+c-a-b-c=16\\
4a-2b+c-a-b-c=21
\end{cases}$
$\begin{cases}
8a+2b=16\\
3a-3b=21
\end{cases}$
Simplify:
$\begin{cases}
4a+b=8\\
a-b=7
\end{cases}$
Add the two equations to eliminate $b$ and determine $a$:
$4a+b+a-b=8+7$
$5a=15$
$a=3$
Substitute the value of $a$ in the Equation $a-b=7$ to determine $b$:
$a-b=7$
$3-b=7$
$b=-4$
Substitute the values of $,b$ is Equation 1 of the given system to find $c$:
$a+b+c=4$
$3+(-4)+c=4$
$-1+c=4$
$c=5$
The system's solution is:
$(3,-4,5)$
The function is:
$y=3x^2-4x+5$
