Find the Equation Given the Roots 1 , 6

$1$ , $6$

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

$y = 0$ at the roots

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

The factor is $x - 1$

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

The factor is $x - 6$

Combine all the factors into a single equation.

$y = (x - 1) (x - 6)$

Multiply all the factors to simplify the equation $y = (x - 1) (x - 6)$ .

Expand $(x - 1) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 6) - 1 (x - 6)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 6 - 1 (x - 6)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 6 - 1 x - 1 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 6 - 1 x - 1 \cdot - 6$

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

$y = x^{2} - 6 \cdot x - 1 x - 1 \cdot - 6$

Rewrite $- 1 x$ as $- x$ .

$y = x^{2} - 6 x - x - 1 \cdot - 6$

Multiply $- 1$ by $- 6$ .

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

Subtract $x$ from $- 6 x$ .

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