Find the Equation Given the Roots 1 , 2- square root of 3

$1$ , $2 - \sqrt{3}$

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 $x = 2 - \sqrt{3}$ was found by solving for $x$ when $x - (2 - \sqrt{3}) = y$ and $y = 0$ .

The factor is $x - 2 + \sqrt{3}$

Step 4

Combine all the factors into a single equation.

$y = (x - 1) (x - 2 + \sqrt{3})$

Step 5

Multiply all the factors to simplify the equation $y = (x - 1) (x - 2 + \sqrt{3})$ .

Expand $(x - 1) (x - 2 + \sqrt{3})$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

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

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

Multiply $- 1$ by $- 2$ .

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

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Subtract $x$ from $- 2 x$ .

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

Step 6