Find the Equation Given the Roots 7 , -1

$7$ , $- 1$

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 $x = 7$ was found by solving for $x$ when $x - (7) = y$ and $y = 0$ .

The factor is $x - 7$

Step 3

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

The factor is $x + 1$

Step 4

Combine all the factors into a single equation.

$y = (x - 7) (x + 1)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 7) (x + 1)$ .

Expand $(x - 7) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$y = x (x + 1) - 7 (x + 1)$

Apply the distributive property.

$y = x \cdot x + x \cdot 1 - 7 (x + 1)$

Apply the distributive property.

$y = x \cdot x + x \cdot 1 - 7 x - 7 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 1 - 7 x - 7 \cdot 1$

Multiply $x$ by $1$ .

$y = x^{2} + x - 7 x - 7 \cdot 1$

Multiply $- 7$ by $1$ .

$y = x^{2} + x - 7 x - 7$

Subtract $7 x$ from $x$ .

$y = x^{2} - 6 x - 7$

Step 6