# Use a system of linear equations to find the quadratic function f(x) = ax^22+bx+c that satisfies the given conditions. Solve the system using matrices. f(-2) = 6, f(1) = -3, f(2) = -14 f(x) =?

Use a system of linear equations to find the quadratic function
$f\left(x\right)=a{x}^{2}2+bx+c$
that satisfies the given conditions. Solve the system using matrices.
f(-2) = 6, f(1) = -3, f(2) = -14
f(x) =?
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krolaniaN
Step 1
$f\left(x\right)=a{x}^{2}+bx+c$
f(-2)=6 , f(1)=-3 , f(2)=-14
f(-2)=6
$⇒4a-2b+c=6\dots \left(1\right)$
$f\left(1\right)=-3$
$⇒a+b+c=-3\dots \left(2\right)$
$f\left(2\right)=-14$
$⇒4a+2b+c=-14\dots \left(3\right)$
Step 2
Eq.(1) + Eq(3)
$⇒8a+2c=-8\dots \left(4\right)$
$\left[2×\text{Eq(2)}\right]+\text{Eq.(1)}$
$⇒6a+3c=0$
$⇒c=-2a\dots \left(5\right)$
From Eq.(4) and Eq.(5)
$8a+2\left(-2a\right)=-8$
$4a=-8$
$a=-2$
$\therefore c=-2×\left(-2\right)=4$
Now,from Eq.(2)
$a+b+c=-3$
$⇒-2+b+4=-3$
$⇒b=-5$
$\therefore a=-2,b=-5,c=4$
$\therefore f\left(x\right)=-2{x}^{2}-5x+4$
Jeffrey Jordon