Find the Equation Given the Roots -5 , 8

$- 5$ , $8$

Step 1

Roots are the points where the graph intercepts with the x-axis $(y = 0)$ .

$y = 0$ at the roots

Step 2

The root at $x = - 5$ was found by solving for $x$ when $x - (- 5) = y$ and $y = 0$ .

The factor is $x + 5$

Step 3

The root at $x = 8$ was found by solving for $x$ when $x - (8) = y$ and $y = 0$ .

The factor is $x - 8$

Step 4

Combine all the factors into a single equation.

$y = (x + 5) (x - 8)$

Step 5

Multiply all the factors to simplify the equation $y = (x + 5) (x - 8)$ .

Expand $(x + 5) (x - 8)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 8) + 5 (x - 8)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 8 + 5 (x - 8)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 8 + 5 x + 5 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 8 + 5 x + 5 \cdot - 8$

Move $- 8$ to the left of $x$ .

$y = x^{2} - 8 \cdot x + 5 x + 5 \cdot - 8$

Multiply $5$ by $- 8$ .

$y = x^{2} - 8 x + 5 x - 40$

Add $- 8 x$ and $5 x$ .

$y = x^{2} - 3 x - 40$

Step 6