Solve for x (x+ square root of 3)^2-(x- square root of 3)^2=8 square root of 3

${(x + \sqrt{3})}^{2} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Simplify ${(x + \sqrt{3})}^{2} - {(x - \sqrt{3})}^{2}$ .

Simplify each term.

Rewrite ${(x + \sqrt{3})}^{2}$ as $(x + \sqrt{3}) (x + \sqrt{3})$ .

$(x + \sqrt{3}) (x + \sqrt{3}) - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Expand $(x + \sqrt{3}) (x + \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$x (x + \sqrt{3}) + \sqrt{3} (x + \sqrt{3}) - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Apply the distributive property.

$x \cdot x + x \sqrt{3} + \sqrt{3} (x + \sqrt{3}) - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Apply the distributive property.

$x \cdot x + x \sqrt{3} + \sqrt{3} x + \sqrt{3} \sqrt{3} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{3} \sqrt{3} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Combine using the product rule for radicals.

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{3 \cdot 3} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Multiply $3$ by $3$ .

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{9} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

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

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{3^{2}} - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Pull terms out from under the radical, assuming positive real numbers.

$x^{2} + x \sqrt{3} + \sqrt{3} x + 3 - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Reorder the factors of $\sqrt{3} x$ .

$x^{2} + x \sqrt{3} + x \sqrt{3} + 3 - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Add $x \sqrt{3}$ and $x \sqrt{3}$ .

$x^{2} + 2 x \sqrt{3} + 3 - {(x - \sqrt{3})}^{2} = 8 \sqrt{3}$

Rewrite ${(x - \sqrt{3})}^{2}$ as $(x - \sqrt{3}) (x - \sqrt{3})$ .

$x^{2} + 2 x \sqrt{3} + 3 - ((x - \sqrt{3}) (x - \sqrt{3})) = 8 \sqrt{3}$

Expand $(x - \sqrt{3}) (x - \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - (x (x - \sqrt{3}) - \sqrt{3} (x - \sqrt{3})) = 8 \sqrt{3}$

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - (x \cdot x + x (- \sqrt{3}) - \sqrt{3} (x - \sqrt{3})) = 8 \sqrt{3}$

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - (x \cdot x + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3})) = 8 \sqrt{3}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3})) = 8 \sqrt{3}$

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

Multiply $- 1$ by $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 1 \sqrt{3} \sqrt{3}) = 8 \sqrt{3}$

Multiply $\sqrt{3}$ by $1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + \sqrt{3} \sqrt{3}) = 8 \sqrt{3}$

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1} \sqrt{3}) = 8 \sqrt{3}$

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1} {\sqrt{3}}^{1}) = 8 \sqrt{3}$

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1 + 1}) = 8 \sqrt{3}$

Add $1$ and $1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{2}) = 8 \sqrt{3}$

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

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {(3^{\frac{1}{2}})}^{2}) = 8 \sqrt{3}$

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{1}{2} \cdot 2}) = 8 \sqrt{3}$

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}}) = 8 \sqrt{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}}) = 8 \sqrt{3}$

Rewrite the expression.

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{1}) = 8 \sqrt{3}$

Evaluate the exponent.

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3) = 8 \sqrt{3}$

Reorder the factors of $- \sqrt{3} x$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - x \sqrt{3} + 3) = 8 \sqrt{3}$

Subtract $x \sqrt{3}$ from $x (- \sqrt{3})$ .

Reorder $x$ and $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} - 1 \cdot x \sqrt{3} - x \sqrt{3} + 3) = 8 \sqrt{3}$

Subtract $x \sqrt{3}$ from $- 1 \cdot x \sqrt{3}$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} - 2 x \sqrt{3} + 3) = 8 \sqrt{3}$

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} - (- 2 x \sqrt{3}) - 1 \cdot 3 = 8 \sqrt{3}$

Simplify.

Multiply $- 2$ by $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} + 2 (x \sqrt{3}) - 1 \cdot 3 = 8 \sqrt{3}$

Multiply $- 1$ by $3$ .

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} + 2 (x \sqrt{3}) - 3 = 8 \sqrt{3}$

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} + 2 x \sqrt{3} - 3 = 8 \sqrt{3}$

Simplify by adding terms.

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

Subtract $x^{2}$ from $x^{2}$ .

$2 x \sqrt{3} + 3 + 0 + 2 x \sqrt{3} - 3 = 8 \sqrt{3}$

Add $2 x \sqrt{3} + 3$ and $0$ .

$2 x \sqrt{3} + 3 + 2 x \sqrt{3} - 3 = 8 \sqrt{3}$

Subtract $3$ from $3$ .

$2 x \sqrt{3} + 2 x \sqrt{3} + 0 = 8 \sqrt{3}$

Add $2 x \sqrt{3} + 2 x \sqrt{3}$ and $0$ .

$2 x \sqrt{3} + 2 x \sqrt{3} = 8 \sqrt{3}$

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

$4 x \sqrt{3} = 8 \sqrt{3}$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 2$

Solve for x (x+ square root of 3)^2-(x- square root of 3)^2=8 square root of 3

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