Find the Equation Given the Roots 1 , 3

$1$ , $3$

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 $x = 3$ was found by solving for $x$ when $x - (3) = y$ and $y = 0$ .

The factor is $x - 3$

Step 4

Combine all the factors into a single equation.

$y = (x - 1) (x - 3)$

Step 5

Multiply all the factors to simplify the equation $y = (x - 1) (x - 3)$ .

Expand $(x - 1) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$y = x (x - 3) - 1 (x - 3)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 3 - 1 (x - 3)$

Apply the distributive property.

$y = x \cdot x + x \cdot - 3 - 1 x - 1 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 3 - 1 x - 1 \cdot - 3$

Move $- 3$ to the left of $x$ .

$y = x^{2} - 3 \cdot x - 1 x - 1 \cdot - 3$

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

$y = x^{2} - 3 x - x - 1 \cdot - 3$

Multiply $- 1$ by $- 3$ .

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

Subtract $x$ from $- 3 x$ .

$y = x^{2} - 4 x + 3$

Step 6