Find the Quadratic Equation Given the Roots square root of 2 , square root of 8

$\sqrt{2}$ , $\sqrt{8}$

Step 1

$x = \sqrt{2}$ and $x = \sqrt{8}$ are the two real distinct solutions for the quadratic equation, which means that $x - \sqrt{2}$ and $x - \sqrt{8}$ are the factors of the quadratic equation.

$(x - \sqrt{2}) (x - \sqrt{8}) = 0$

Step 2

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(x - \sqrt{2}) (x - \sqrt{4 (2)}) = 0$

Rewrite $4$ as $2^{2}$ .

$(x - \sqrt{2}) (x - \sqrt{2^{2} \cdot 2}) = 0$

Pull terms out from under the radical.

$(x - \sqrt{2}) (x - (2 \sqrt{2})) = 0$

Multiply $2$ by $- 1$ .

$(x - \sqrt{2}) (x - 2 \sqrt{2}) = 0$

Step 3

Expand $(x - \sqrt{2}) (x - 2 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$x (x - 2 \sqrt{2}) - \sqrt{2} (x - 2 \sqrt{2}) = 0$

Apply the distributive property.

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

Apply the distributive property.

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

Step 4

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x - \sqrt{2} (- 2 \sqrt{2}) = 0$

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

Multiply $- 2$ by $- 1$ .

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \sqrt{2} \sqrt{2} = 0$

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 (\sqrt{2} \sqrt{2}) = 0$

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 (\sqrt{2} \sqrt{2}) = 0$

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 {\sqrt{2}}^{1 + 1} = 0$

Add $1$ and $1$ .

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 {\sqrt{2}}^{2} = 0$

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

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 {(2^{\frac{1}{2}})}^{2} = 0$

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \cdot 2^{\frac{1}{2} \cdot 2} = 0$

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

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \cdot 2^{\frac{2}{2}} = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \cdot 2^{\frac{2}{2}} = 0$

Rewrite the expression.

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \cdot 2 = 0$

Evaluate the exponent.

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 2 \cdot 2 = 0$

Multiply $2$ by $2$ .

$x^{2} + x (- 2 \sqrt{2}) - \sqrt{2} x + 4 = 0$

Reorder the factors of $- \sqrt{2} x$ .

$x^{2} + x (- 2 \sqrt{2}) - x \sqrt{2} + 4 = 0$

Subtract $x \sqrt{2}$ from $x (- 2 \sqrt{2})$ .

Reorder $x$ and $- 2$ .

$x^{2} - 2 \cdot (x \sqrt{2}) - x \sqrt{2} + 4 = 0$

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

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

Step 5

The standard quadratic equation using the given set of solutions ${\sqrt{2}, \sqrt{8}}$ is $y = x^{2} - 3 x \sqrt{2} + 4$ .

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

Step 6

Find the Quadratic Equation Given the Roots square root of 2 , square root of 8

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