# How do you find the equation for a parabola if you have (3,0), (5,0) and (0, 15)?

Jun 14, 2016

$y = {x}^{2} - 8 x + 15$

#### Explanation:

Standard form of equation of a parabola is $y = a {x}^{2} + b x + c$

As it passes through points $\left(3 , 0\right)$, $\left(5.0\right)$ and $\left(0 , 15\right)$, each of these points satisfies the equation of parabola and hence

$0 = a \cdot 9 + b \cdot 3 + c$ or $9 a + 3 b + c = 0$ ........(A)
$0 = a \cdot 25 + b \cdot 5 + c$ or $25 a + 5 b + c = 0$ ........(B)
and $15 = a \cdot 0 + b \cdot 0 + c$ or $c = 15$ ........(C)

Now putting (C) in (A) and (B), we get\

$9 a + 3 b = - 15$ or $3 a + b = - 5$ and .........(1)

$25 a + 5 b = - 15$ or $5 a + b = - 3$ .........(2)

Subtracting (1) from (2), we get $2 a = 2$ or $a = 1$

and hence $b = - 5 - 3 \cdot 1 = - 8$

Hence equation of parabola is

$y = {x}^{2} - 8 x + 15$ and it appears as shown below

graph{x^2-8x+15 [-5.5, 14.5, -2, 8.84]}