Find the Equation Given the Roots 3 , 2+ square root of 6

$3$ , $2 + \sqrt{6}$

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

The factor is $x - 3$

Step 3

The root at $x = 2 + \sqrt{6}$ was found by solving for $x$ when $x - (2 + \sqrt{6}) = y$ and $y = 0$ .

The factor is $x - 2 - \sqrt{6}$

Step 4

Combine all the factors into a single equation.

$y = (x - 3) (x - 2 - \sqrt{6})$

Step 5

Multiply all the factors to simplify the equation $y = (x - 3) (x - 2 - \sqrt{6})$ .

Expand $(x - 3) (x - 2 - \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 2 + x (- \sqrt{6}) - 3 x - 3 \cdot - 2 - 3 (- \sqrt{6})$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 2 + x (- \sqrt{6}) - 3 x - 3 \cdot - 2 - 3 (- \sqrt{6})$

Move $- 2$ to the left of $x$ .

$y = x^{2} - 2 \cdot x + x (- \sqrt{6}) - 3 x - 3 \cdot - 2 - 3 (- \sqrt{6})$

Multiply $- 3$ by $- 2$ .

$y = x^{2} - 2 x + x (- \sqrt{6}) - 3 x + 6 - 3 (- \sqrt{6})$

Multiply $- 1$ by $- 3$ .

$y = x^{2} - 2 x + x (- \sqrt{6}) - 3 x + 6 + 3 \sqrt{6}$

Simplify by adding terms.

Subtract $3 x$ from $- 2 x$ .

$y = x^{2} + x (- \sqrt{6}) - 5 x + 6 + 3 \sqrt{6}$

Reorder factors in $x^{2} + x (- \sqrt{6}) - 5 x + 6 + 3 \sqrt{6}$ .

$y = x^{2} - x \sqrt{6} - 5 x + 6 + 3 \sqrt{6}$

Step 6