Find the Equation Given the Roots -1 , 5

$- 1$ , $5$

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 $x = - 1$ was found by solving for $x$ when $x - (- 1) = y$ and $y = 0$ .

The factor is $x + 1$

Step 3

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

The factor is $x - 5$

Step 4

Combine all the factors into a single equation.

$y = (x + 1) (x - 5)$

Step 5

Multiply all the factors to simplify the equation $y = (x + 1) (x - 5)$ .

Expand $(x + 1) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 5) + 1 (x - 5)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 5 + 1 (x - 5)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 5 + 1 x + 1 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 5 + 1 x + 1 \cdot - 5$

Move $- 5$ to the left of $x$ .

$y = x^{2} - 5 \cdot x + 1 x + 1 \cdot - 5$

Multiply $x$ by $1$ .

$y = x^{2} - 5 x + x + 1 \cdot - 5$

Multiply $- 5$ by $1$ .

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

Add $- 5 x$ and $x$ .

$y = x^{2} - 4 x - 5$

Step 6