Algebra Examples

$x^{\frac{3}{2}} = 8$

Step 1

Subtract $8$ from both sides of the equation.

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

Step 2

Rewrite $x^{\frac{3}{2}}$ as ${(x^{\frac{1}{2}})}^{3}$ .

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

Step 3

Rewrite $8$ as $2^{3}$ .

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

Step 4

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x^{\frac{1}{2}}$ and $b = 2$ .

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

Step 5

Simplify.

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Step 5.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 5.1.1

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

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

Step 5.1.2

Cancel the common factor of $2$ .

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Step 5.1.2.1

Cancel the common factor.

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

Step 5.1.2.2

Rewrite the expression.

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

Step 5.2

Simplify.

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

Step 5.3

Move $2$ to the left of $x^{\frac{1}{2}}$ .

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

Step 5.4

Raise $2$ to the power of $2$ .

$(x^{\frac{1}{2}} - 2) (x + 2 x^{\frac{1}{2}} + 4) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

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

$x + 2 x^{\frac{1}{2}} + 4 = 0$

Step 7

Set $x^{\frac{1}{2}} - 2$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{\frac{1}{2}} - 2$ equal to $0$ .

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

Step 7.2

Solve $x^{\frac{1}{2}} - 2 = 0$ for $x$ .

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Step 7.2.1

Add $2$ to both sides of the equation.

$x^{\frac{1}{2}} = 2$

Step 7.2.2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 7.2.3.1.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 7.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 2^{2}$

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 2^{2}$

Step 7.2.3.1.1.2

Simplify.

$x = 2^{2}$

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

Raise $2$ to the power of $2$ .

$x = 4$

Step 8

Set $x + 2 x^{\frac{1}{2}} + 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x + 2 x^{\frac{1}{2}} + 4$ equal to $0$ .

$x + 2 x^{\frac{1}{2}} + 4 = 0$

Step 8.2

Solve $x + 2 x^{\frac{1}{2}} + 4 = 0$ for $x$ .

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Step 8.2.1

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

${(x^{\frac{1}{2}})}^{2} + 2 x^{\frac{1}{2}} + 4$

Step 8.2.2

Substitute $u$ for $x^{\frac{1}{2}}$ .

${(u)}^{2} + 2 (u) + 4 = 0$

Step 8.2.3

Solve for $u$ .

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Step 8.2.3.1

Multiply $2$ by $u$ .

$u^{2} + 2 u + 4 = 0$

Step 8.2.3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 8.2.3.3

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $u$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1}$

Step 8.2.3.4

Simplify.

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Step 8.2.3.4.1

Simplify the numerator.

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Step 8.2.3.4.1.1

Raise $2$ to the power of $2$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.2.3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.4.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.2.3.4.1.3

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.2.3.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.2.3.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.2.3.4.1.7.1

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.2.3.4.1.7.2

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.2.3.4.1.8

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.2.3.4.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.2.3.4.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.2.3.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Step 8.2.3.5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.2.3.5.1

Simplify the numerator.

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Step 8.2.3.5.1.1

Raise $2$ to the power of $2$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.2.3.5.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.5.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.2.3.5.1.3

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.2.3.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.2.3.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.2.3.5.1.7.1

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.2.3.5.1.7.2

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.2.3.5.1.8

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.2.3.5.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.2.3.5.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.2.3.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Step 8.2.3.5.4

Change the $\pm$ to $+$ .

$u = - 1 + i \sqrt{3}$

Step 8.2.3.6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 8.2.3.6.1

Simplify the numerator.

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Step 8.2.3.6.1.1

Raise $2$ to the power of $2$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.6.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 8.2.3.6.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 8.2.3.6.1.2.2

Multiply $- 4$ by $4$ .

$u = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 8.2.3.6.1.3

Subtract $16$ from $4$ .

$u = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 8.2.3.6.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$u = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 8.2.3.6.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$u = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.6.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$u = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 8.2.3.6.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 8.2.3.6.1.7.1

Factor $4$ out of $12$ .

$u = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 8.2.3.6.1.7.2

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

$u = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 8.2.3.6.1.8

Pull terms out from under the radical.

$u = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 8.2.3.6.1.9

Move $2$ to the left of $i$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 8.2.3.6.2

Multiply $2$ by $1$ .

$u = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 8.2.3.6.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$u = - 1 \pm i \sqrt{3}$

Step 8.2.3.6.4

Change the $\pm$ to $-$ .

$u = - 1 - i \sqrt{3}$

Step 8.2.3.7

The final answer is the combination of both solutions.

$u = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 8.2.4

Substitute $x$ for $u$ .

$x^{\frac{1}{2}} = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 8.2.5

Solve for $x^{\frac{1}{2}} = - 1 + i \sqrt{3}$ for $x$ .

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Step 8.2.5.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 8.2.5.2

Simplify the exponent.

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Step 8.2.5.2.1

Simplify the left side.

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Step 8.2.5.2.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.5.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.5.2.1.1.1.1

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

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

Step 8.2.5.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 8.2.5.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.5.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.5.2.1.1.2

Simplify.

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

Step 8.2.5.2.2

Simplify the right side.

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Step 8.2.5.2.2.1

Simplify ${(- 1 + i \sqrt{3})}^{2}$ .

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Step 8.2.5.2.2.1.1

Rewrite ${(- 1 + i \sqrt{3})}^{2}$ as $(- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$ .

$x = (- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$

Step 8.2.5.2.2.1.2

Expand $(- 1 + i \sqrt{3}) (- 1 + i \sqrt{3})$ using the FOIL Method.

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Step 8.2.5.2.2.1.2.1

Apply the distributive property.

$x = - 1 (- 1 + i \sqrt{3}) + i \sqrt{3} (- 1 + i \sqrt{3})$

Step 8.2.5.2.2.1.2.2

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (i \sqrt{3}) + i \sqrt{3} (- 1 + i \sqrt{3})$

Step 8.2.5.2.2.1.2.3

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Step 8.2.5.2.2.1.3

Simplify and combine like terms.

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Step 8.2.5.2.2.1.3.1

Simplify each term.

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Step 8.2.5.2.2.1.3.1.1

Multiply $- 1$ by $- 1$ .

$x = 1 - 1 (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Step 8.2.5.2.2.1.3.1.2

Rewrite $- 1 (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = 1 - (i \sqrt{3}) + i \sqrt{3} \cdot - 1 + i \sqrt{3} (i \sqrt{3})$

Step 8.2.5.2.2.1.3.1.3

Move $- 1$ to the left of $i \sqrt{3}$ .

$x = 1 - i \sqrt{3} - 1 \cdot (i \sqrt{3}) + i \sqrt{3} (i \sqrt{3})$

Step 8.2.5.2.2.1.3.1.4

Rewrite $- 1 (i \sqrt{3})$ as $- (i \sqrt{3})$ .

$x = 1 - i \sqrt{3} - (i \sqrt{3}) + i \sqrt{3} (i \sqrt{3})$

Step 8.2.5.2.2.1.3.1.5

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

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Step 8.2.5.2.2.1.3.1.5.1

Raise $i$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1} i \sqrt{3} \sqrt{3}$

Step 8.2.5.2.2.1.3.1.5.2

Raise $i$ to the power of $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1} i^{1} \sqrt{3} \sqrt{3}$

Step 8.2.5.2.2.1.3.1.5.3

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

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{1 + 1} \sqrt{3} \sqrt{3}$

Step 8.2.5.2.2.1.3.1.5.4

Add $1$ and $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} \sqrt{3} \sqrt{3}$

Step 8.2.5.2.2.1.3.1.5.5

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

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

Step 8.2.5.2.2.1.3.1.5.6

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

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

Step 8.2.5.2.2.1.3.1.5.7

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

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} {\sqrt{3}}^{1 + 1}$

Step 8.2.5.2.2.1.3.1.5.8

Add $1$ and $1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} + i^{2} {\sqrt{3}}^{2}$

Step 8.2.5.2.2.1.3.1.6

Rewrite $i^{2}$ as $- 1$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 {\sqrt{3}}^{2}$

Step 8.2.5.2.2.1.3.1.7

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

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Step 8.2.5.2.2.1.3.1.7.1

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

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

Step 8.2.5.2.2.1.3.1.7.2

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

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 8.2.5.2.2.1.3.1.7.3

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

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}$

Step 8.2.5.2.2.1.3.1.7.4

Cancel the common factor of $2$ .

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Step 8.2.5.2.2.1.3.1.7.4.1

Cancel the common factor.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{\frac{2}{2}}$

Step 8.2.5.2.2.1.3.1.7.4.2

Rewrite the expression.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3^{1}$

Step 8.2.5.2.2.1.3.1.7.5

Evaluate the exponent.

$x = 1 - i \sqrt{3} - i \sqrt{3} - 1 \cdot 3$

Step 8.2.5.2.2.1.3.1.8

Multiply $- 1$ by $3$ .

$x = 1 - i \sqrt{3} - i \sqrt{3} - 3$

Step 8.2.5.2.2.1.3.2

Subtract $3$ from $1$ .

$x = - i \sqrt{3} - i \sqrt{3} - 2$

Step 8.2.5.2.2.1.3.3

Subtract $i \sqrt{3}$ from $- i \sqrt{3}$ .

$x = - 2 i \sqrt{3} - 2$

Step 8.2.5.2.2.1.4

Reorder $- 2 i \sqrt{3}$ and $- 2$ .

$x = - 2 - 2 i \sqrt{3}$

Step 8.2.6

Solve for $x^{\frac{1}{2}} = - 1 - i \sqrt{3}$ for $x$ .

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Step 8.2.6.1

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Step 8.2.6.2

Simplify the exponent.

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Step 8.2.6.2.1

Simplify the left side.

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Step 8.2.6.2.1.1

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.6.2.1.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

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Step 8.2.6.2.1.1.1.1

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

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

Step 8.2.6.2.1.1.1.2

Cancel the common factor of $2$ .

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Step 8.2.6.2.1.1.1.2.1

Cancel the common factor.

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

Step 8.2.6.2.1.1.1.2.2

Rewrite the expression.

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

Step 8.2.6.2.1.1.2

Simplify.

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

Step 8.2.6.2.2

Simplify the right side.

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Step 8.2.6.2.2.1

Simplify ${(- 1 - i \sqrt{3})}^{2}$ .

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Step 8.2.6.2.2.1.1

Rewrite ${(- 1 - i \sqrt{3})}^{2}$ as $(- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$ .

$x = (- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$

Step 8.2.6.2.2.1.2

Expand $(- 1 - i \sqrt{3}) (- 1 - i \sqrt{3})$ using the FOIL Method.

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Step 8.2.6.2.2.1.2.1

Apply the distributive property.

$x = - 1 (- 1 - i \sqrt{3}) - i \sqrt{3} (- 1 - i \sqrt{3})$

Step 8.2.6.2.2.1.2.2

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (- i \sqrt{3}) - i \sqrt{3} (- 1 - i \sqrt{3})$

Step 8.2.6.2.2.1.2.3

Apply the distributive property.

$x = - 1 \cdot - 1 - 1 (- i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3

Simplify and combine like terms.

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Step 8.2.6.2.2.1.3.1

Simplify each term.

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Step 8.2.6.2.2.1.3.1.1

Multiply $- 1$ by $- 1$ .

$x = 1 - 1 (- i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.2

Multiply $- 1 (- i \sqrt{3})$ .

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Step 8.2.6.2.2.1.3.1.2.1

Multiply $- 1$ by $- 1$ .

$x = 1 + 1 (i \sqrt{3}) - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.2.2

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

$x = 1 + \sqrt{3} i - i \sqrt{3} \cdot - 1 - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.3

Multiply $- i \sqrt{3} \cdot - 1$ .

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Step 8.2.6.2.2.1.3.1.3.1

Multiply $- 1$ by $- 1$ .

$x = 1 + \sqrt{3} i + 1 i \sqrt{3} - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.3.2

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

$x = 1 + \sqrt{3} i + \sqrt{3} i - i \sqrt{3} (- i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.4

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

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Step 8.2.6.2.2.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + 1 i \sqrt{3} (i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.4.2

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + \sqrt{3} i (i \sqrt{3})$

Step 8.2.6.2.2.1.3.1.4.3

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} \sqrt{3} i i$

Step 8.2.6.2.2.1.3.1.4.4

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1} {\sqrt{3}}^{1} i i$

Step 8.2.6.2.2.1.3.1.4.5

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{1 + 1} i i$

Step 8.2.6.2.2.1.3.1.4.6

Add $1$ and $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i i$

Step 8.2.6.2.2.1.3.1.4.7

Raise $i$ to the power of $1$ .

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

Step 8.2.6.2.2.1.3.1.4.8

Raise $i$ to the power of $1$ .

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

Step 8.2.6.2.2.1.3.1.4.9

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{1 + 1}$

Step 8.2.6.2.2.1.3.1.4.10

Add $1$ and $1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + {\sqrt{3}}^{2} i^{2}$

Step 8.2.6.2.2.1.3.1.5

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

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Step 8.2.6.2.2.1.3.1.5.1

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

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

Step 8.2.6.2.2.1.3.1.5.2

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{1}{2} \cdot 2} i^{2}$

Step 8.2.6.2.2.1.3.1.5.3

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

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Step 8.2.6.2.2.1.3.1.5.4

Cancel the common factor of $2$ .

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Step 8.2.6.2.2.1.3.1.5.4.1

Cancel the common factor.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{\frac{2}{2}} i^{2}$

Step 8.2.6.2.2.1.3.1.5.4.2

Rewrite the expression.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3^{1} i^{2}$

Step 8.2.6.2.2.1.3.1.5.5

Evaluate the exponent.

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3 i^{2}$

Step 8.2.6.2.2.1.3.1.6

Rewrite $i^{2}$ as $- 1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i + 3 \cdot - 1$

Step 8.2.6.2.2.1.3.1.7

Multiply $3$ by $- 1$ .

$x = 1 + \sqrt{3} i + \sqrt{3} i - 3$

Step 8.2.6.2.2.1.3.2

Subtract $3$ from $1$ .

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

Step 8.2.6.2.2.1.3.3

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

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

Step 8.2.7

List all of the solutions.

$x = - 2 - 2 i \sqrt{3}, - 2 + 2 \sqrt{3} i$

Step 9

The final solution is all the values that make $(x^{\frac{1}{2}} - 2) (x + 2 x^{\frac{1}{2}} + 4) = 0$ true.

$x = 4, - 2 - 2 i \sqrt{3}, - 2 + 2 \sqrt{3} i$