Find the Equation Given the Roots 7+ square root of 3 , 2- square root of 6

$7 + \sqrt{3}$ , $2 - \sqrt{6}$

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 $x = 7 + \sqrt{3}$ was found by solving for $x$ when $x - (7 + \sqrt{3}) = y$ and $y = 0$ .

The factor is $x - 7 - \sqrt{3}$

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.

$y = (x - 7 - \sqrt{3}) (x - 2 + \sqrt{6})$

Step 5

Multiply all the factors to simplify the equation $y = (x - 7 - \sqrt{3}) (x - 2 + \sqrt{6})$ .

Expand $(x - 7 - \sqrt{3}) (x - 2 + \sqrt{6})$ by multiplying each term in the first expression by each term in the second expression.

$y = x \cdot x + x \cdot - 2 + x \sqrt{6} - 7 x - 7 \cdot - 2 - 7 \sqrt{6} - \sqrt{3} x - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{6}$

Simplify terms.

Simplify each term.

Multiply $x$ by $x$ .

$y = x^{2} + x \cdot - 2 + x \sqrt{6} - 7 x - 7 \cdot - 2 - 7 \sqrt{6} - \sqrt{3} x - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{6}$

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

$y = x^{2} - 2 \cdot x + x \sqrt{6} - 7 x - 7 \cdot - 2 - 7 \sqrt{6} - \sqrt{3} x - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{6}$

Multiply $- 7$ by $- 2$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{6}$

Multiply $- 2$ by $- 1$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - \sqrt{3} \sqrt{6}$

Multiply $- \sqrt{3} \sqrt{6}$ .

Combine using the product rule for radicals.

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - \sqrt{6 \cdot 3}$

Multiply $6$ by $3$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - \sqrt{18}$

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - (3 \sqrt{2})$

Multiply $3$ by $- 1$ .

$y = x^{2} - 2 x + x \sqrt{6} - 7 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - 3 \sqrt{2}$

Simplify by adding terms.

Subtract $7 x$ from $- 2 x$ .

$y = x^{2} + x \sqrt{6} - 9 x + 14 - 7 \sqrt{6} - \sqrt{3} x + 2 \sqrt{3} - 3 \sqrt{2}$

Reorder.

Move $- 7 \sqrt{6}$ .

$y = x^{2} + x \sqrt{6} - 9 x + 14 - \sqrt{3} x - 7 \sqrt{6} + 2 \sqrt{3} - 3 \sqrt{2}$

Move $14$ .

$y = x^{2} + x \sqrt{6} - 9 x - \sqrt{3} x + 14 - 7 \sqrt{6} + 2 \sqrt{3} - 3 \sqrt{2}$

Step 6

Find the Equation Given the Roots 7+ square root of 3 , 2- square root of 6

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