Find the Quadratic Equation 4 square root of 2 , -4 square root of 2

$4 \sqrt{2}$ , $- 4 \sqrt{2}$

Step 1

$x = 4 \sqrt{2}$ and $x = - 4 \sqrt{2}$ are the two real distinct solutions for the quadratic equation, which means that $x - 4 \sqrt{2}$ and $x + 4 \sqrt{2}$ are the factors of the quadratic equation.

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

Step 2

Expand $(x - 4 \sqrt{2}) (x + 4 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify terms.

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

Reorder the factors in the terms $x (4 \sqrt{2})$ and $- 4 \sqrt{2} x$ .

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

Subtract $4 x \sqrt{2}$ from $4 x \sqrt{2}$ .

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

Add $x \cdot x$ and $0$ .

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

Simplify each term.

Multiply $x$ by $x$ .

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

Multiply $- 4 \sqrt{2} (4 \sqrt{2})$ .

Multiply $4$ by $- 4$ .

$x^{2} - 16 \sqrt{2} \sqrt{2} = 0$

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

$x^{2} - 16 (\sqrt{2} \sqrt{2}) = 0$

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

$x^{2} - 16 (\sqrt{2} \sqrt{2}) = 0$

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

$x^{2} - 16 {\sqrt{2}}^{1 + 1} = 0$

Add $1$ and $1$ .

$x^{2} - 16 {\sqrt{2}}^{2} = 0$

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

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

$x^{2} - 16 {(2^{\frac{1}{2}})}^{2} = 0$

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

$x^{2} - 16 \cdot 2^{\frac{1}{2} \cdot 2} = 0$

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

$x^{2} - 16 \cdot 2^{\frac{2}{2}} = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} - 16 \cdot 2^{\frac{2}{2}} = 0$

Rewrite the expression.

$x^{2} - 16 \cdot 2 = 0$

Evaluate the exponent.

$x^{2} - 16 \cdot 2 = 0$

Multiply $- 16$ by $2$ .

$x^{2} - 32 = 0$

Step 4

The standard quadratic equation using the given set of solutions ${4 \sqrt{2}, - 4 \sqrt{2}}$ is $y = x^{2} - 32$ .

$y = x^{2} - 32$

Step 5

Find the Quadratic Equation 4 square root of 2 , -4 square root of 2

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