Find the Equation Given the Roots -7 , 6

$- 7$ , $6$

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 $x = 6$ was found by solving for $x$ when $x - (6) = y$ and $y = 0$ .

The factor is $x - 6$

Step 4

Combine all the factors into a single equation.

$y = (x + 7) (x - 6)$

Step 5

Multiply all the factors to simplify the equation $y = (x + 7) (x - 6)$ .

Expand $(x + 7) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 6) + 7 (x - 6)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 6 + 7 (x - 6)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 6 + 7 x + 7 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 6 + 7 x + 7 \cdot - 6$

Move $- 6$ to the left of $x$ .

$y = x^{2} - 6 \cdot x + 7 x + 7 \cdot - 6$

Multiply $7$ by $- 6$ .

$y = x^{2} - 6 x + 7 x - 42$

Add $- 6 x$ and $7 x$ .

$y = x^{2} + x - 42$

Step 6