Algebra Examples

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

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $8 \sqrt{3}$ from both sides of the equation.

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

Step 1.2

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

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

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

Step 1.2.1.2

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

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Step 1.2.1.2.1

Apply the distributive property.

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

Step 1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \sqrt{3} + \sqrt{3} (x + \sqrt{3}) - {(x - \sqrt{3})}^{2} - 8 \sqrt{3} = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \sqrt{3} + \sqrt{3} x + \sqrt{3} \sqrt{3} - {(x - \sqrt{3})}^{2} - 8 \sqrt{3} = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.2.1.3.1.2

Combine using the product rule for radicals.

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{3 \cdot 3} - {(x - \sqrt{3})}^{2} - 8 \sqrt{3} = 0$

Step 1.2.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} + x \sqrt{3} + \sqrt{3} x + \sqrt{9} - {(x - \sqrt{3})}^{2} - 8 \sqrt{3} = 0$

Step 1.2.1.3.1.4

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

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

Step 1.2.1.3.1.5

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

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

Step 1.2.1.3.2

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

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

Step 1.2.1.3.3

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

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

Step 1.2.1.4

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

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

Step 1.2.1.5

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

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Step 1.2.1.5.1

Apply the distributive property.

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

Step 1.2.1.5.2

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - (x \cdot x + x (- \sqrt{3}) - \sqrt{3} (x - \sqrt{3})) - 8 \sqrt{3} = 0$

Step 1.2.1.5.3

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - (x \cdot x + x (- \sqrt{3}) - \sqrt{3} x - \sqrt{3} (- \sqrt{3})) - 8 \sqrt{3} = 0$

Step 1.2.1.6

Simplify and combine like terms.

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Step 1.2.1.6.1

Simplify each term.

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Step 1.2.1.6.1.1

Multiply $x$ by $x$ .

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

Step 1.2.1.6.1.2

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

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Step 1.2.1.6.1.2.1

Multiply $- 1$ by $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 1 \sqrt{3} \sqrt{3}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.2.2

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

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

Step 1.2.1.6.1.2.3

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1} \sqrt{3}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.2.4

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1} {\sqrt{3}}^{1}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.2.5

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {\sqrt{3}}^{1 + 1}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.2.6

Add $1$ and $1$ .

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

Step 1.2.1.6.1.3

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

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Step 1.2.1.6.1.3.1

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + {(3^{\frac{1}{2}})}^{2}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.3.2

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{1}{2} \cdot 2}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.3.3

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

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.3.4

Cancel the common factor of $2$ .

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Step 1.2.1.6.1.3.4.1

Cancel the common factor.

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{\frac{2}{2}}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.3.4.2

Rewrite the expression.

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} + x (- \sqrt{3}) - \sqrt{3} x + 3^{1}) - 8 \sqrt{3} = 0$

Step 1.2.1.6.1.3.5

Evaluate the exponent.

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

Step 1.2.1.6.2

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

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

Step 1.2.1.6.3

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

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Step 1.2.1.6.3.1

Reorder $x$ and $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - (x^{2} - 1 \cdot x \sqrt{3} - x \sqrt{3} + 3) - 8 \sqrt{3} = 0$

Step 1.2.1.6.3.2

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

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

Step 1.2.1.7

Apply the distributive property.

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} - (- 2 x \sqrt{3}) - 1 \cdot 3 - 8 \sqrt{3} = 0$

Step 1.2.1.8

Simplify.

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Step 1.2.1.8.1

Multiply $- 2$ by $- 1$ .

$x^{2} + 2 x \sqrt{3} + 3 - x^{2} + 2 (x \sqrt{3}) - 1 \cdot 3 - 8 \sqrt{3} = 0$

Step 1.2.1.8.2

Multiply $- 1$ by $3$ .

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

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

Step 1.2.2

Combine the opposite terms in $x^{2} + 2 x \sqrt{3} + 3 - x^{2} + 2 x \sqrt{3} - 3 - 8 \sqrt{3}$ .

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Step 1.2.2.1

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

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Subtract $3$ from $3$ .

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

Step 1.2.2.4

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

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

Step 1.2.3

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

$4 x \sqrt{3} - 8 \sqrt{3} = 0$

Step 2

Add $8 \sqrt{3}$ to both sides of the equation.

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

Step 3

Divide each term in $4 x \sqrt{3} = 8 \sqrt{3}$ by $4 \sqrt{3}$ and simplify.

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Step 3.1

Divide each term in $4 x \sqrt{3} = 8 \sqrt{3}$ by $4 \sqrt{3}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

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

Step 3.2.2

Cancel the common factor of $\sqrt{3}$ .

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Step 3.2.2.1

Cancel the common factor.

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

Step 3.2.2.2

Divide $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $8$ and $4$ .

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Step 3.3.1.1

Factor $4$ out of $8 \sqrt{3}$ .

$x = \frac{4 (2 \sqrt{3})}{4 \sqrt{3}}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $4$ out of $4 \sqrt{3}$ .

$x = \frac{4 (2 \sqrt{3})}{4 (\sqrt{3})}$

Step 3.3.1.2.2

Cancel the common factor.

$x = \frac{4 (2 \sqrt{3})}{4 \sqrt{3}}$

Step 3.3.1.2.3

Rewrite the expression.

$x = \frac{2 \sqrt{3}}{\sqrt{3}}$

Step 3.3.2

Cancel the common factor of $\sqrt{3}$ .

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Step 3.3.2.1

Cancel the common factor.

$x = \frac{2 \sqrt{3}}{\sqrt{3}}$

Step 3.3.2.2

Divide $2$ by $1$ .

$x = 2$