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

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

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

$y = 0$ at the roots

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 $x - 2 - \sqrt{3}$

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 $x - 2 + \sqrt{3}$

Combine all the factors into a single equation.

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

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

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

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

Simplify terms.

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Combine the opposite terms in $x \cdot x + x \cdot - 2 + x \sqrt{3} - 2 x - 2 \cdot - 2 - 2 \sqrt{3} - \sqrt{3} x - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{3}$ .

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Reorder the factors in the terms $x \sqrt{3}$ and $- \sqrt{3} x$ .

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

Subtract $x \sqrt{3}$ from $x \sqrt{3}$ .

$y = x \cdot x + x \cdot - 2 + 0 - 2 x - 2 \cdot - 2 - 2 \sqrt{3} - \sqrt{3} \cdot - 2 - \sqrt{3} \sqrt{3}$

Add $x \cdot x + x \cdot - 2$ and $0$ .

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

Simplify each term.

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Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2$ by $- 1$ .

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

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

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Raise $\sqrt{3}$ to the power of $1$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - (\sqrt{3} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - (\sqrt{3} \sqrt{3})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - {(3^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - 3^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

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

Evaluate the exponent.

$y = x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

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

Simplify by adding terms.

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Combine the opposite terms in $x^{2} - 2 x - 2 x + 4 - 2 \sqrt{3} + 2 \sqrt{3} - 3$ .

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Add $- 2 \sqrt{3}$ and $2 \sqrt{3}$ .

$y = x^{2} - 2 x - 2 x + 4 + 0 - 3$

Add $x^{2} - 2 x - 2 x + 4$ and $0$ .

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

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

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

Subtract $3$ from $4$ .

$y = x^{2} - 4 x + 1$

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

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