Algebra Examples

${(\sqrt{20 - x})}^{3} = 343$

Step 1

Solve for $\sqrt{20 - x}$ .

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Step 1.1

Subtract $343$ from both sides of the equation.

${\sqrt{20 - x}}^{3} - 343 = 0$

Step 1.2

Factor the left side of the equation.

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Step 1.2.1

Rewrite $343$ as $7^{3}$ .

${\sqrt{20 - x}}^{3} - 7^{3} = 0$

Step 1.2.2

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 $a = \sqrt{20 - x}$ and $b = 7$ .

$(\sqrt{20 - x} - 7) ({\sqrt{20 - x}}^{2} + \sqrt{20 - x} \cdot 7 + 7^{2}) = 0$

Step 1.2.3

Simplify.

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Step 1.2.3.1

Move $7$ to the left of $\sqrt{20 - x}$ .

$(\sqrt{20 - x} - 7) ({\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 7^{2}) = 0$

Step 1.2.3.2

Raise $7$ to the power of $2$ .

$(\sqrt{20 - x} - 7) ({\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49) = 0$

Step 1.3

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

$\sqrt{20 - x} - 7 = 0$

${\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49 = 0$

Step 1.4

Set $\sqrt{20 - x} - 7$ equal to $0$ and solve for $\sqrt{20 - x}$ .

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Step 1.4.1

Set $\sqrt{20 - x} - 7$ equal to $0$ .

$\sqrt{20 - x} - 7 = 0$

Step 1.4.2

Add $7$ to both sides of the equation.

$\sqrt{20 - x} = 7$

Step 1.5

Set ${\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49$ equal to $0$ and solve for $\sqrt{20 - x}$ .

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Step 1.5.1

Set ${\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49$ equal to $0$ .

${\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49 = 0$

Step 1.5.2

Solve ${\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49 = 0$ for $\sqrt{20 - x}$ .

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Step 1.5.2.1

Use the quadratic formula to find the solutions.

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

Step 1.5.2.2

Substitute the values $a = 1$ , $b = 7$ , and $c = 49$ into the quadratic formula and solve for $\sqrt{20 - x}$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot 49)}}{2 \cdot 1}$

Step 1.5.2.3

Simplify.

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Step 1.5.2.3.1

Simplify the numerator.

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Step 1.5.2.3.1.1

Raise $7$ to the power of $2$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}$

Step 1.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 49$ .

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Step 1.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{49 - 4 \cdot 49}}{2 \cdot 1}$

Step 1.5.2.3.1.2.2

Multiply $- 4$ by $49$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{49 - 196}}{2 \cdot 1}$

Step 1.5.2.3.1.3

Subtract $196$ from $49$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{- 147}}{2 \cdot 1}$

Step 1.5.2.3.1.4

Rewrite $- 147$ as $- 1 (147)$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Step 1.5.2.3.1.5

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$\sqrt{20 - x} = \frac{- 7 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

Step 1.5.2.3.1.6

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

$\sqrt{20 - x} = \frac{- 7 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Step 1.5.2.3.1.7

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 1.5.2.3.1.7.1

Factor $49$ out of $147$ .

$\sqrt{20 - x} = \frac{- 7 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Step 1.5.2.3.1.7.2

Rewrite $49$ as $7^{2}$ .

$\sqrt{20 - x} = \frac{- 7 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Step 1.5.2.3.1.8

Pull terms out from under the radical.

$\sqrt{20 - x} = \frac{- 7 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Step 1.5.2.3.1.9

Move $7$ to the left of $i$ .

$\sqrt{20 - x} = \frac{- 7 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Step 1.5.2.3.2

Multiply $2$ by $1$ .

$\sqrt{20 - x} = \frac{- 7 \pm 7 i \sqrt{3}}{2}$

Step 1.5.2.4

The final answer is the combination of both solutions.

$\sqrt{20 - x} = - \frac{7 - 7 i \sqrt{3}}{2}, - \frac{7 + 7 i \sqrt{3}}{2}$

Step 1.6

The final solution is all the values that make $(\sqrt{20 - x} - 7) ({\sqrt{20 - x}}^{2} + 7 \sqrt{20 - x} + 49) = 0$ true.

$\sqrt{20 - x} = 7, - \frac{7 - 7 i \sqrt{3}}{2}, - \frac{7 + 7 i \sqrt{3}}{2}$

Step 2

Solve for $x$ in $\sqrt{20 - x} = 7$ .

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Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{20 - x}}^{2} = 7^{2}$

Step 2.2

Simplify each side of the equation.

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 - x}$ as ${(20 - x)}^{\frac{1}{2}}$ .

${({(20 - x)}^{\frac{1}{2}})}^{2} = 7^{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Simplify ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1

Multiply the exponents in ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.2.1.1.1

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

${(20 - x)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.2.1.1.2.1

Cancel the common factor.

${(20 - x)}^{\frac{1}{2} \cdot 2} = 7^{2}$

Step 2.2.2.1.1.2.2

Rewrite the expression.

${(20 - x)}^{1} = 7^{2}$

Step 2.2.2.1.2

Simplify.

$20 - x = 7^{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raise $7$ to the power of $2$ .

$20 - x = 49$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1.1

Subtract $20$ from both sides of the equation.

$- x = 49 - 20$

Step 2.3.1.2

Subtract $20$ from $49$ .

$- x = 29$

Step 2.3.2

Divide each term in $- x = 29$ by $- 1$ and simplify.

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Step 2.3.2.1

Divide each term in $- x = 29$ by $- 1$ .

$\frac{- x}{- 1} = \frac{29}{- 1}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{29}{- 1}$

Step 2.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{29}{- 1}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Divide $29$ by $- 1$ .

$x = - 29$

Step 3

Solve for $x$ in $\sqrt{20 - x} = - \frac{7 - 7 i \sqrt{3}}{2}$ .

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Step 3.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{20 - x}}^{2} = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2

Simplify each side of the equation.

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Step 3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 - x}$ as ${(20 - x)}^{\frac{1}{2}}$ .

${({(20 - x)}^{\frac{1}{2}})}^{2} = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Simplify ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1

Multiply the exponents in ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.2.1.1.1

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

${(20 - x)}^{\frac{1}{2} \cdot 2} = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.2.1

Cancel the common factor.

${(20 - x)}^{\frac{1}{2} \cdot 2} = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.2.1.1.2.2

Rewrite the expression.

${(20 - x)}^{1} = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.2.1.2

Simplify.

$20 - x = {(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Simplify ${(- \frac{7 - 7 i \sqrt{3}}{2})}^{2}$ .

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Step 3.2.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.2.3.1.1.1

Apply the product rule to $- \frac{7 - 7 i \sqrt{3}}{2}$ .

$20 - x = {(- 1)}^{2} {(\frac{7 - 7 i \sqrt{3}}{2})}^{2}$

Step 3.2.3.1.1.2

Apply the product rule to $\frac{7 - 7 i \sqrt{3}}{2}$ .

$20 - x = {(- 1)}^{2} \frac{{(7 - 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 3.2.3.1.2

Simplify the expression.

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Step 3.2.3.1.2.1

Raise $- 1$ to the power of $2$ .

$20 - x = 1 \frac{{(7 - 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 3.2.3.1.2.2

Multiply $\frac{{(7 - 7 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

$20 - x = \frac{{(7 - 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 3.2.3.1.2.3

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

$20 - x = \frac{{(7 - 7 i \sqrt{3})}^{2}}{4}$

Step 3.2.3.1.2.4

Rewrite ${(7 - 7 i \sqrt{3})}^{2}$ as $(7 - 7 i \sqrt{3}) (7 - 7 i \sqrt{3})$ .

$20 - x = \frac{(7 - 7 i \sqrt{3}) (7 - 7 i \sqrt{3})}{4}$

Step 3.2.3.1.3

Expand $(7 - 7 i \sqrt{3}) (7 - 7 i \sqrt{3})$ using the FOIL Method.

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Step 3.2.3.1.3.1

Apply the distributive property.

$20 - x = \frac{7 (7 - 7 i \sqrt{3}) - 7 i \sqrt{3} (7 - 7 i \sqrt{3})}{4}$

Step 3.2.3.1.3.2

Apply the distributive property.

$20 - x = \frac{7 \cdot 7 + 7 (- 7 i \sqrt{3}) - 7 i \sqrt{3} (7 - 7 i \sqrt{3})}{4}$

Step 3.2.3.1.3.3

Apply the distributive property.

$20 - x = \frac{7 \cdot 7 + 7 (- 7 i \sqrt{3}) - 7 i \sqrt{3} \cdot 7 - 7 i \sqrt{3} (- 7 i \sqrt{3})}{4}$

Step 3.2.3.1.4

Simplify and combine like terms.

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Step 3.2.3.1.4.1

Simplify each term.

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Step 3.2.3.1.4.1.1

Multiply $7$ by $7$ .

$20 - x = \frac{49 + 7 (- 7 i \sqrt{3}) - 7 i \sqrt{3} \cdot 7 - 7 i \sqrt{3} (- 7 i \sqrt{3})}{4}$

Step 3.2.3.1.4.1.2

Multiply $- 7$ by $7$ .

$20 - x = \frac{49 - 49 (i \sqrt{3}) - 7 i \sqrt{3} \cdot 7 - 7 i \sqrt{3} (- 7 i \sqrt{3})}{4}$

Step 3.2.3.1.4.1.3

Multiply $7$ by $- 7$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 7 i \sqrt{3} (- 7 i \sqrt{3})}{4}$

Step 3.2.3.1.4.1.4

Multiply $- 7 i \sqrt{3} (- 7 i \sqrt{3})$ .

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Step 3.2.3.1.4.1.4.1

Multiply $- 7$ by $- 7$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i \sqrt{3} (i \sqrt{3})}{4}$

Step 3.2.3.1.4.1.4.2

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 3.2.3.1.4.1.4.3

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 3.2.3.1.4.1.4.4

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 3.2.3.1.4.1.4.5

Add $1$ and $1$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 3.2.3.1.4.1.4.6

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 3.2.3.1.4.1.4.7

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 3.2.3.1.4.1.4.8

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 3.2.3.1.4.1.4.9

Add $1$ and $1$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 i^{2} {\sqrt{3}}^{2}}{4}$

Step 3.2.3.1.4.1.5

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} + 49 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 3.2.3.1.4.1.6

Multiply $49$ by $- 1$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 {\sqrt{3}}^{2}}{4}$

Step 3.2.3.1.4.1.7

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

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Step 3.2.3.1.4.1.7.1

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 3.2.3.1.4.1.7.2

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 3.2.3.1.4.1.7.3

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

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 \cdot 3^{\frac{2}{2}}}{4}$

Step 3.2.3.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 3.2.3.1.4.1.7.4.1

Cancel the common factor.

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 \cdot 3^{\frac{2}{2}}}{4}$

Step 3.2.3.1.4.1.7.4.2

Rewrite the expression.

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 \cdot 3^{1}}{4}$

Step 3.2.3.1.4.1.7.5

Evaluate the exponent.

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 49 \cdot 3}{4}$

Step 3.2.3.1.4.1.8

Multiply $- 49$ by $3$ .

$20 - x = \frac{49 - 49 i \sqrt{3} - 49 i \sqrt{3} - 147}{4}$

Step 3.2.3.1.4.2

Subtract $147$ from $49$ .

$20 - x = \frac{- 49 i \sqrt{3} - 49 i \sqrt{3} - 98}{4}$

Step 3.2.3.1.4.3

Subtract $49 i \sqrt{3}$ from $- 49 i \sqrt{3}$ .

$20 - x = \frac{- 98 i \sqrt{3} - 98}{4}$

Step 3.2.3.1.5

Reorder $- 98 i \sqrt{3}$ and $- 98$ .

$20 - x = \frac{- 98 - 98 i \sqrt{3}}{4}$

Step 3.2.3.1.6

Cancel the common factor of $- 98 - 98 i \sqrt{3}$ and $4$ .

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Step 3.2.3.1.6.1

Factor $2$ out of $- 98$ .

$20 - x = \frac{2 (- 49) - 98 i \sqrt{3}}{4}$

Step 3.2.3.1.6.2

Factor $2$ out of $- 98 i \sqrt{3}$ .

$20 - x = \frac{2 (- 49) + 2 (- 49 i \sqrt{3})}{4}$

Step 3.2.3.1.6.3

Factor $2$ out of $2 (- 49) + 2 (- 49 i \sqrt{3})$ .

$20 - x = \frac{2 (- 49 - 49 i \sqrt{3})}{4}$

Step 3.2.3.1.6.4

Cancel the common factors.

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Step 3.2.3.1.6.4.1

Factor $2$ out of $4$ .

$20 - x = \frac{2 (- 49 - 49 i \sqrt{3})}{2 \cdot 2}$

Step 3.2.3.1.6.4.2

Cancel the common factor.

$20 - x = \frac{2 (- 49 - 49 i \sqrt{3})}{2 \cdot 2}$

Step 3.2.3.1.6.4.3

Rewrite the expression.

$20 - x = \frac{- 49 - 49 i \sqrt{3}}{2}$

Step 3.2.3.1.7

Rewrite $- 49$ as $- 1 (49)$ .

$20 - x = \frac{- 1 (49) - 49 i \sqrt{3}}{2}$

Step 3.2.3.1.8

Factor $- 1$ out of $- 49 i \sqrt{3}$ .

$20 - x = \frac{- 1 (49) - (49 i \sqrt{3})}{2}$

Step 3.2.3.1.9

Factor $- 1$ out of $- 1 (49) - (49 i \sqrt{3})$ .

$20 - x = \frac{- 1 (49 + 49 i \sqrt{3})}{2}$

Step 3.2.3.1.10

Move the negative in front of the fraction.

$20 - x = - \frac{49 + 49 i \sqrt{3}}{2}$

Step 3.3

Solve for $x$ .

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Step 3.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.3.1.1

Subtract $20$ from both sides of the equation.

$- x = - \frac{49 + 49 i \sqrt{3}}{2} - 20$

Step 3.3.1.2

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x = - \frac{49 + 49 i \sqrt{3}}{2} - 20 \cdot \frac{2}{2}$

Step 3.3.1.3

Combine $- 20$ and $\frac{2}{2}$ .

$- x = - \frac{49 + 49 i \sqrt{3}}{2} + \frac{- 20 \cdot 2}{2}$

Step 3.3.1.4

Combine the numerators over the common denominator.

$- x = \frac{- 89 - 49 i \sqrt{3}}{2}$

Step 3.3.1.5

Rewrite $- 89$ as $- 1 (89)$ .

$- x = \frac{- 1 (89) - 49 i \sqrt{3}}{2}$

Step 3.3.1.6

Factor $- 1$ out of $- 49 i \sqrt{3}$ .

$- x = \frac{- 1 (89) - (49 i \sqrt{3})}{2}$

Step 3.3.1.7

Factor $- 1$ out of $- 1 (89) - (49 i \sqrt{3})$ .

$- x = \frac{- 1 (89 + 49 i \sqrt{3})}{2}$

Step 3.3.1.8

Move the negative in front of the fraction.

$- x = - \frac{89 + 49 i \sqrt{3}}{2}$

Step 3.3.2

Divide each term in $- x = - \frac{89 + 49 i \sqrt{3}}{2}$ by $- 1$ and simplify.

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Step 3.3.2.1

Divide each term in $- x = - \frac{89 + 49 i \sqrt{3}}{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- \frac{89 + 49 i \sqrt{3}}{2}}{- 1}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- \frac{89 + 49 i \sqrt{3}}{2}}{- 1}$

Step 3.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{89 + 49 i \sqrt{3}}{2}}{- 1}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{89 + 49 i \sqrt{3}}{2}}{1}$

Step 3.3.2.3.2

Divide $\frac{89 + 49 i \sqrt{3}}{2}$ by $1$ .

$x = \frac{89 + 49 i \sqrt{3}}{2}$

Step 4

Solve for $x$ in $\sqrt{20 - x} = - \frac{7 + 7 i \sqrt{3}}{2}$ .

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Step 4.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{20 - x}}^{2} = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2

Simplify each side of the equation.

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{20 - x}$ as ${(20 - x)}^{\frac{1}{2}}$ .

${({(20 - x)}^{\frac{1}{2}})}^{2} = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Simplify ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1

Multiply the exponents in ${({(20 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 4.2.2.1.1.1

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

${(20 - x)}^{\frac{1}{2} \cdot 2} = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.2.1

Cancel the common factor.

${(20 - x)}^{\frac{1}{2} \cdot 2} = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.2.1.1.2.2

Rewrite the expression.

${(20 - x)}^{1} = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.2.1.2

Simplify.

$20 - x = {(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify ${(- \frac{7 + 7 i \sqrt{3}}{2})}^{2}$ .

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Step 4.2.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.3.1.1.1

Apply the product rule to $- \frac{7 + 7 i \sqrt{3}}{2}$ .

$20 - x = {(- 1)}^{2} {(\frac{7 + 7 i \sqrt{3}}{2})}^{2}$

Step 4.2.3.1.1.2

Apply the product rule to $\frac{7 + 7 i \sqrt{3}}{2}$ .

$20 - x = {(- 1)}^{2} \frac{{(7 + 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 4.2.3.1.2

Simplify the expression.

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Step 4.2.3.1.2.1

Raise $- 1$ to the power of $2$ .

$20 - x = 1 \frac{{(7 + 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 4.2.3.1.2.2

Multiply $\frac{{(7 + 7 i \sqrt{3})}^{2}}{2^{2}}$ by $1$ .

$20 - x = \frac{{(7 + 7 i \sqrt{3})}^{2}}{2^{2}}$

Step 4.2.3.1.2.3

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

$20 - x = \frac{{(7 + 7 i \sqrt{3})}^{2}}{4}$

Step 4.2.3.1.2.4

Rewrite ${(7 + 7 i \sqrt{3})}^{2}$ as $(7 + 7 i \sqrt{3}) (7 + 7 i \sqrt{3})$ .

$20 - x = \frac{(7 + 7 i \sqrt{3}) (7 + 7 i \sqrt{3})}{4}$

Step 4.2.3.1.3

Expand $(7 + 7 i \sqrt{3}) (7 + 7 i \sqrt{3})$ using the FOIL Method.

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Step 4.2.3.1.3.1

Apply the distributive property.

$20 - x = \frac{7 (7 + 7 i \sqrt{3}) + 7 i \sqrt{3} (7 + 7 i \sqrt{3})}{4}$

Step 4.2.3.1.3.2

Apply the distributive property.

$20 - x = \frac{7 \cdot 7 + 7 (7 i \sqrt{3}) + 7 i \sqrt{3} (7 + 7 i \sqrt{3})}{4}$

Step 4.2.3.1.3.3

Apply the distributive property.

$20 - x = \frac{7 \cdot 7 + 7 (7 i \sqrt{3}) + 7 i \sqrt{3} \cdot 7 + 7 i \sqrt{3} (7 i \sqrt{3})}{4}$

Step 4.2.3.1.4

Simplify and combine like terms.

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Step 4.2.3.1.4.1

Simplify each term.

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Step 4.2.3.1.4.1.1

Multiply $7$ by $7$ .

$20 - x = \frac{49 + 7 (7 i \sqrt{3}) + 7 i \sqrt{3} \cdot 7 + 7 i \sqrt{3} (7 i \sqrt{3})}{4}$

Step 4.2.3.1.4.1.2

Multiply $7$ by $7$ .

$20 - x = \frac{49 + 49 (i \sqrt{3}) + 7 i \sqrt{3} \cdot 7 + 7 i \sqrt{3} (7 i \sqrt{3})}{4}$

Step 4.2.3.1.4.1.3

Multiply $7$ by $7$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 7 i \sqrt{3} (7 i \sqrt{3})}{4}$

Step 4.2.3.1.4.1.4

Multiply $7 i \sqrt{3} (7 i \sqrt{3})$ .

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Step 4.2.3.1.4.1.4.1

Multiply $7$ by $7$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i \sqrt{3} (i \sqrt{3})}{4}$

Step 4.2.3.1.4.1.4.2

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 (i^{1} i) \sqrt{3} \sqrt{3}}{4}$

Step 4.2.3.1.4.1.4.3

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 (i^{1} i^{1}) \sqrt{3} \sqrt{3}}{4}$

Step 4.2.3.1.4.1.4.4

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{1 + 1} \sqrt{3} \sqrt{3}}{4}$

Step 4.2.3.1.4.1.4.5

Add $1$ and $1$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{2} \sqrt{3} \sqrt{3}}{4}$

Step 4.2.3.1.4.1.4.6

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{2} ({\sqrt{3}}^{1} \sqrt{3})}{4}$

Step 4.2.3.1.4.1.4.7

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{4}$

Step 4.2.3.1.4.1.4.8

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{2} {\sqrt{3}}^{1 + 1}}{4}$

Step 4.2.3.1.4.1.4.9

Add $1$ and $1$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 i^{2} {\sqrt{3}}^{2}}{4}$

Step 4.2.3.1.4.1.5

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} + 49 \cdot - 1 {\sqrt{3}}^{2}}{4}$

Step 4.2.3.1.4.1.6

Multiply $49$ by $- 1$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 {\sqrt{3}}^{2}}{4}$

Step 4.2.3.1.4.1.7

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

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Step 4.2.3.1.4.1.7.1

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 {(3^{\frac{1}{2}})}^{2}}{4}$

Step 4.2.3.1.4.1.7.2

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 \cdot 3^{\frac{1}{2} \cdot 2}}{4}$

Step 4.2.3.1.4.1.7.3

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

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.2.3.1.4.1.7.4

Cancel the common factor of $2$ .

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Step 4.2.3.1.4.1.7.4.1

Cancel the common factor.

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 \cdot 3^{\frac{2}{2}}}{4}$

Step 4.2.3.1.4.1.7.4.2

Rewrite the expression.

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 \cdot 3^{1}}{4}$

Step 4.2.3.1.4.1.7.5

Evaluate the exponent.

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 49 \cdot 3}{4}$

Step 4.2.3.1.4.1.8

Multiply $- 49$ by $3$ .

$20 - x = \frac{49 + 49 i \sqrt{3} + 49 i \sqrt{3} - 147}{4}$

Step 4.2.3.1.4.2

Subtract $147$ from $49$ .

$20 - x = \frac{49 i \sqrt{3} + 49 i \sqrt{3} - 98}{4}$

Step 4.2.3.1.4.3

Add $49 i \sqrt{3}$ and $49 i \sqrt{3}$ .

$20 - x = \frac{98 i \sqrt{3} - 98}{4}$

Step 4.2.3.1.5

Reorder $98 i \sqrt{3}$ and $- 98$ .

$20 - x = \frac{- 98 + 98 i \sqrt{3}}{4}$

Step 4.2.3.1.6

Cancel the common factor of $- 98 + 98 i \sqrt{3}$ and $4$ .

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Step 4.2.3.1.6.1

Factor $2$ out of $- 98$ .

$20 - x = \frac{2 (- 49) + 98 i \sqrt{3}}{4}$

Step 4.2.3.1.6.2

Factor $2$ out of $98 i \sqrt{3}$ .

$20 - x = \frac{2 (- 49) + 2 (49 i \sqrt{3})}{4}$

Step 4.2.3.1.6.3

Factor $2$ out of $2 (- 49) + 2 (49 i \sqrt{3})$ .

$20 - x = \frac{2 (- 49 + 49 i \sqrt{3})}{4}$

Step 4.2.3.1.6.4

Cancel the common factors.

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Step 4.2.3.1.6.4.1

Factor $2$ out of $4$ .

$20 - x = \frac{2 (- 49 + 49 i \sqrt{3})}{2 \cdot 2}$

Step 4.2.3.1.6.4.2

Cancel the common factor.

$20 - x = \frac{2 (- 49 + 49 i \sqrt{3})}{2 \cdot 2}$

Step 4.2.3.1.6.4.3

Rewrite the expression.

$20 - x = \frac{- 49 + 49 i \sqrt{3}}{2}$

Step 4.2.3.1.7

Rewrite $- 49$ as $- 1 (49)$ .

$20 - x = \frac{- 1 (49) + 49 i \sqrt{3}}{2}$

Step 4.2.3.1.8

Factor $- 1$ out of $49 i \sqrt{3}$ .

$20 - x = \frac{- 1 (49) - (- 49 i \sqrt{3})}{2}$

Step 4.2.3.1.9

Factor $- 1$ out of $- 1 (49) - (- 49 i \sqrt{3})$ .

$20 - x = \frac{- 1 (49 - 49 i \sqrt{3})}{2}$

Step 4.2.3.1.10

Move the negative in front of the fraction.

$20 - x = - \frac{49 - 49 i \sqrt{3}}{2}$

Step 4.3

Solve for $x$ .

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Step 4.3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.3.1.1

Subtract $20$ from both sides of the equation.

$- x = - \frac{49 - 49 i \sqrt{3}}{2} - 20$

Step 4.3.1.2

To write $- 20$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- x = - \frac{49 - 49 i \sqrt{3}}{2} - 20 \cdot \frac{2}{2}$

Step 4.3.1.3

Combine $- 20$ and $\frac{2}{2}$ .

$- x = - \frac{49 - 49 i \sqrt{3}}{2} + \frac{- 20 \cdot 2}{2}$

Step 4.3.1.4

Combine the numerators over the common denominator.

$- x = \frac{- 89 + 49 i \sqrt{3}}{2}$

Step 4.3.1.5

Rewrite $- 89$ as $- 1 (89)$ .

$- x = \frac{- 1 (89) + 49 i \sqrt{3}}{2}$

Step 4.3.1.6

Factor $- 1$ out of $49 i \sqrt{3}$ .

$- x = \frac{- 1 (89) - (- 49 i \sqrt{3})}{2}$

Step 4.3.1.7

Factor $- 1$ out of $- 1 (89) - (- 49 i \sqrt{3})$ .

$- x = \frac{- 1 (89 - 49 i \sqrt{3})}{2}$

Step 4.3.1.8

Move the negative in front of the fraction.

$- x = - \frac{89 - 49 i \sqrt{3}}{2}$

Step 4.3.2

Divide each term in $- x = - \frac{89 - 49 i \sqrt{3}}{2}$ by $- 1$ and simplify.

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Step 4.3.2.1

Divide each term in $- x = - \frac{89 - 49 i \sqrt{3}}{2}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- \frac{89 - 49 i \sqrt{3}}{2}}{- 1}$

Step 4.3.2.2

Simplify the left side.

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Step 4.3.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- \frac{89 - 49 i \sqrt{3}}{2}}{- 1}$

Step 4.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{- \frac{89 - 49 i \sqrt{3}}{2}}{- 1}$

Step 4.3.2.3

Simplify the right side.

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Step 4.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{\frac{89 - 49 i \sqrt{3}}{2}}{1}$

Step 4.3.2.3.2

Divide $\frac{89 - 49 i \sqrt{3}}{2}$ by $1$ .

$x = \frac{89 - 49 i \sqrt{3}}{2}$

Step 5

List all of the solutions.

$x = - 29, \frac{89 + 49 i \sqrt{3}}{2}, \frac{89 - 49 i \sqrt{3}}{2}$