Find the Equation Given the Roots 6 , -9

$6$ , $- 9$

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

The factor is $x - 6$

Step 3

The root at $x = - 9$ was found by solving for $x$ when $x - (- 9) = y$ and $y = 0$ .

The factor is $x + 9$

Step 4

Combine all the factors into a single equation.

$y = (x - 6) (x + 9)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 6) (x + 9)$ .

Expand $(x - 6) (x + 9)$ using the FOIL Method.

Apply the distributive property.

$y = x (x + 9) - 6 (x + 9)$

Apply the distributive property.

$y = x \cdot x + x \cdot 9 - 6 (x + 9)$

Apply the distributive property.

$y = x \cdot x + x \cdot 9 - 6 x - 6 \cdot 9$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot 9 - 6 x - 6 \cdot 9$

Move $9$ to the left of $x$ .

$y = x^{2} + 9 \cdot x - 6 x - 6 \cdot 9$

Multiply $- 6$ by $9$ .

$y = x^{2} + 9 x - 6 x - 54$

Subtract $6 x$ from $9 x$ .

$y = x^{2} + 3 x - 54$

Step 6