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

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

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

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

Step 3

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

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

Step 4

Combine all the factors into a single equation.

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

Step 5

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

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

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

Simplify terms.

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

Reorder the factors in the terms $x \sqrt{2}$ and $- \sqrt{2} x$ .

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

Subtract $x \sqrt{2}$ from $x \sqrt{2}$ .

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

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

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

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 2$ by $- 2$ .

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- \sqrt{2} \sqrt{2}$ .

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $2$ .

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

Simplify by adding terms.

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

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

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

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

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

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

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

Subtract $2$ from $4$ .

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

Step 6

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

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