facas9

2021-10-24

What is the equation of a quadratic function whose graph passes through $\left(2,0\right)$, $\left(5,0\right)$, and $\left(3,-8\right)$? Write your equation in standard form.

okomgcae

Consider the given:
The graph passes through $\left(2,0\right)$, $\left(5,0\right)$ and $\left(3,-8\right)$.
Let the equation is $y=a{x}^{2}+bx+c$.
Consider the point $\left(2,0\right)$,
$a{\left(2\right)}^{2}+b\left(2\right)+c=0$
$4a+2b+c=0$ ... ... (1)
Consider the point $\left(5,0\right)$,
$a{\left(5\right)}^{2}+b\left(5\right)+c=0$
$25a+5b+c=0$ ... ... (2)
Consider the point $\left(3,-8\right)$,
$a{\left(3\right)}^{2}+b\left(3\right)+c=-8$
$9a+3b+c=-8$ ... ... (3)
From equation (1) and (2),
$4a+2b+c=25a+5b+c$
$4a+2b=25a+5b$
$a–25a=5b–2b$
$-7a=b$
From equation (1),
$4a+2b+c=0$
$4a+2\left(-7a\right)+c=0$
$4a–14a+c=0$
$-10a+c=0$
$c=10a$
Substitute the value of b and c in equation (3).
$9a+3b+c=-8$
$9a+3\left(-7a\right)+\left(10a\right)=-8$
$9a–21a+10a=-8$
$-21a+19a=-8$
$-2a=-8$
$a=4$
$b=-7a$
$b=-7\left(4\right)$
$b=-28$
$c=10a$
$c=10\left(4\right)$
$c=40$
Hence, The equation of quadratic function will be:
$y=4{x}^{2}–28x+40$

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