Algebra Examples

$a^{2} + 2 b^{2} = 10$ $3 a^{2} - b^{2} = 9$

Step 1

Solve for $a$ in $a^{2} + 2 b^{2} = 10$ .

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Step 1.1

Subtract $2 b^{2}$ from both sides of the equation.

$a^{2} = 10 - 2 b^{2}$

$3 a^{2} - b^{2} = 9$

Step 1.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{10 - 2 b^{2}}$

$3 a^{2} - b^{2} = 9$

Step 1.3

Factor $2$ out of $10 - 2 b^{2}$ .

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Step 1.3.1

Factor $2$ out of $10$ .

$a = \pm \sqrt{2 (5) - 2 b^{2}}$

$3 a^{2} - b^{2} = 9$

Step 1.3.2

Factor $2$ out of $- 2 b^{2}$ .

$a = \pm \sqrt{2 (5) + 2 (- b^{2})}$

$3 a^{2} - b^{2} = 9$

Step 1.3.3

Factor $2$ out of $2 (5) + 2 (- b^{2})$ .

$a = \pm \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

$a = \pm \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

Step 1.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.4.1

First, use the positive value of the $\pm$ to find the first solution.

$a = \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

Step 1.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

Step 1.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = \sqrt{2 (5 - b^{2})}$

$a = - \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$a = - \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$a = - \sqrt{2 (5 - b^{2})}$

$3 a^{2} - b^{2} = 9$

Step 2

Solve the system $a = \sqrt{2 (5 - b^{2})}, 3 a^{2} - b^{2} = 9$ .

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Step 2.1

Replace all occurrences of $a$ with $\sqrt{2 (5 - b^{2})}$ in each equation.

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Step 2.1.1

Replace all occurrences of $a$ in $3 a^{2} - b^{2} = 9$ with $\sqrt{2 (5 - b^{2})}$ .

$3 {(\sqrt{2 (5 - b^{2})})}^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $3 {(\sqrt{2 (5 - b^{2})})}^{2} - b^{2}$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

Rewrite ${\sqrt{2 (5 - b^{2})}}^{2}$ as $2 (5 - b^{2})$ .

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Step 2.1.2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (5 - b^{2})}$ as ${(2 (5 - b^{2}))}^{\frac{1}{2}}$ .

$3 {({(2 (5 - b^{2}))}^{\frac{1}{2}})}^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.1.2

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

$3 {(2 (5 - b^{2}))}^{\frac{1}{2} \cdot 2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.1.3

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

$3 {(2 (5 - b^{2}))}^{\frac{2}{2}} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.1.4

Cancel the common factor of $2$ .

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Step 2.1.2.1.1.1.4.1

Cancel the common factor.

$3 {(2 (5 - b^{2}))}^{\frac{2}{2}} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.1.4.2

Rewrite the expression.

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.1.5

Simplify.

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.2

Apply the distributive property.

$3 (2 \cdot 5 + 2 (- b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.3

Multiply $2$ by $5$ .

$3 (10 + 2 (- b^{2})) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.4

Multiply $- 1$ by $2$ .

$3 (10 - 2 b^{2}) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.5

Apply the distributive property.

$3 \cdot 10 + 3 (- 2 b^{2}) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.6

Multiply $3$ by $10$ .

$30 + 3 (- 2 b^{2}) - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.1.7

Multiply $- 2$ by $3$ .

$30 - 6 b^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$30 - 6 b^{2} - b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.1.2.1.2

Subtract $b^{2}$ from $- 6 b^{2}$ .

$30 - 7 b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2

Solve for $b$ in $30 - 7 b^{2} = 9$ .

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Step 2.2.1

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

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Step 2.2.1.1

Subtract $30$ from both sides of the equation.

$- 7 b^{2} = 9 - 30$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.1.2

Subtract $30$ from $9$ .

$- 7 b^{2} = - 21$

$a = \sqrt{2 (5 - b^{2})}$

$- 7 b^{2} = - 21$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.2

Divide each term in $- 7 b^{2} = - 21$ by $- 7$ and simplify.

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Step 2.2.2.1

Divide each term in $- 7 b^{2} = - 21$ by $- 7$ .

$\frac{- 7 b^{2}}{- 7} = \frac{- 21}{- 7}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 2.2.2.2.1.1

Cancel the common factor.

$\frac{- 7 b^{2}}{- 7} = \frac{- 21}{- 7}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.2.2.1.2

Divide $b^{2}$ by $1$ .

$b^{2} = \frac{- 21}{- 7}$

$a = \sqrt{2 (5 - b^{2})}$

$b^{2} = \frac{- 21}{- 7}$

$a = \sqrt{2 (5 - b^{2})}$

$b^{2} = \frac{- 21}{- 7}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 21$ by $- 7$ .

$b^{2} = 3$

$a = \sqrt{2 (5 - b^{2})}$

$b^{2} = 3$

$a = \sqrt{2 (5 - b^{2})}$

$b^{2} = 3$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$b = \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$b = - \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$b = \sqrt{3}, - \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

$b = \sqrt{3}, - \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

$b = \sqrt{3}, - \sqrt{3}$

$a = \sqrt{2 (5 - b^{2})}$

Step 2.3

Replace all occurrences of $b$ with $\sqrt{3}$ in each equation.

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Step 2.3.1

Replace all occurrences of $b$ in $a = \sqrt{2 (5 - b^{2})}$ with $\sqrt{3}$ .

$a = \sqrt{2 (5 - {(\sqrt{3})}^{2})}$

$b = \sqrt{3}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Simplify $\sqrt{2 (5 - {(\sqrt{3})}^{2})}$ .

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Step 2.3.2.1.1

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

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Step 2.3.2.1.1.1

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

$a = \sqrt{2 (5 - {(3^{\frac{1}{2}})}^{2})}$

$b = \sqrt{3}$

Step 2.3.2.1.1.2

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

$a = \sqrt{2 (5 - 3^{\frac{1}{2} \cdot 2})}$

$b = \sqrt{3}$

Step 2.3.2.1.1.3

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

$a = \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = \sqrt{3}$

Step 2.3.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.3.2.1.1.4.1

Cancel the common factor.

$a = \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = \sqrt{3}$

Step 2.3.2.1.1.4.2

Rewrite the expression.

$a = \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

$a = \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

Step 2.3.2.1.1.5

Evaluate the exponent.

$a = \sqrt{2 (5 - 1 \cdot 3)}$

$b = \sqrt{3}$

$a = \sqrt{2 (5 - 1 \cdot 3)}$

$b = \sqrt{3}$

Step 2.3.2.1.2

Simplify the expression.

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Step 2.3.2.1.2.1

Multiply $- 1$ by $3$ .

$a = \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

Step 2.3.2.1.2.2

Subtract $3$ from $5$ .

$a = \sqrt{2 \cdot 2}$

$b = \sqrt{3}$

Step 2.3.2.1.2.3

Multiply $2$ by $2$ .

$a = \sqrt{4}$

$b = \sqrt{3}$

Step 2.3.2.1.2.4

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

$a = \sqrt{2^{2}}$

$b = \sqrt{3}$

$a = \sqrt{2^{2}}$

$b = \sqrt{3}$

Step 2.3.2.1.3

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

$a = 2$

$b = \sqrt{3}$

$a = 2$

$b = \sqrt{3}$

$a = 2$

$b = \sqrt{3}$

$a = 2$

$b = \sqrt{3}$

Step 2.4

Replace all occurrences of $b$ with $- \sqrt{3}$ in each equation.

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Step 2.4.1

Replace all occurrences of $b$ in $a = \sqrt{2 (5 - b^{2})}$ with $- \sqrt{3}$ .

$a = \sqrt{2 (5 - {(- \sqrt{3})}^{2})}$

$b = - \sqrt{3}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $\sqrt{2 (5 - {(- \sqrt{3})}^{2})}$ .

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Step 2.4.2.1.1

Apply the product rule to $- \sqrt{3}$ .

$a = \sqrt{2 (5 - ({(- 1)}^{2} {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 2.4.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 2.4.2.1.2.1

Move ${(- 1)}^{2}$ .

$a = \sqrt{2 (5 + {(- 1)}^{2} \cdot (- 1 {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 2.4.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 2.4.2.1.2.2.1

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

$a = \sqrt{2 (5 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 2.4.2.1.2.2.2

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

$a = \sqrt{2 (5 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

$a = \sqrt{2 (5 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 2.4.2.1.2.3

Add $2$ and $1$ .

$a = \sqrt{2 (5 + {(- 1)}^{3} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

$a = \sqrt{2 (5 + {(- 1)}^{3} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 2.4.2.1.3

Raise $- 1$ to the power of $3$ .

$a = \sqrt{2 (5 - {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 2.4.2.1.4

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

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Step 2.4.2.1.4.1

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

$a = \sqrt{2 (5 - {(3^{\frac{1}{2}})}^{2})}$

$b = - \sqrt{3}$

Step 2.4.2.1.4.2

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

$a = \sqrt{2 (5 - 3^{\frac{1}{2} \cdot 2})}$

$b = - \sqrt{3}$

Step 2.4.2.1.4.3

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

$a = \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = - \sqrt{3}$

Step 2.4.2.1.4.4

Cancel the common factor of $2$ .

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Step 2.4.2.1.4.4.1

Cancel the common factor.

$a = \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = - \sqrt{3}$

Step 2.4.2.1.4.4.2

Rewrite the expression.

$a = \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

$a = \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

Step 2.4.2.1.4.5

Evaluate the exponent.

$a = \sqrt{2 (5 - 1 \cdot 3)}$

$b = - \sqrt{3}$

$a = \sqrt{2 (5 - 1 \cdot 3)}$

$b = - \sqrt{3}$

Step 2.4.2.1.5

Simplify the expression.

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Step 2.4.2.1.5.1

Multiply $- 1$ by $3$ .

$a = \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

Step 2.4.2.1.5.2

Subtract $3$ from $5$ .

$a = \sqrt{2 \cdot 2}$

$b = - \sqrt{3}$

Step 2.4.2.1.5.3

Multiply $2$ by $2$ .

$a = \sqrt{4}$

$b = - \sqrt{3}$

Step 2.4.2.1.5.4

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

$a = \sqrt{2^{2}}$

$b = - \sqrt{3}$

$a = \sqrt{2^{2}}$

$b = - \sqrt{3}$

Step 2.4.2.1.6

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

$a = 2$

$b = - \sqrt{3}$

$a = 2$

$b = - \sqrt{3}$

$a = 2$

$b = - \sqrt{3}$

$a = 2$

$b = - \sqrt{3}$

$a = 2$

$b = - \sqrt{3}$

Step 3

Solve the system $a = - \sqrt{2 (5 - b^{2})}, 3 a^{2} - b^{2} = 9$ .

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Step 3.1

Replace all occurrences of $a$ with $- \sqrt{2 (5 - b^{2})}$ in each equation.

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Step 3.1.1

Replace all occurrences of $a$ in $3 a^{2} - b^{2} = 9$ with $- \sqrt{2 (5 - b^{2})}$ .

$3 {(- \sqrt{2 (5 - b^{2})})}^{2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Simplify $3 {(- \sqrt{2 (5 - b^{2})})}^{2} - b^{2}$ .

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Step 3.1.2.1.1

Simplify each term.

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Step 3.1.2.1.1.1

Apply the product rule to $- \sqrt{2 (5 - b^{2})}$ .

$3 ({(- 1)}^{2} {\sqrt{2 (5 - b^{2})}}^{2}) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.2

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

$3 (1 {\sqrt{2 (5 - b^{2})}}^{2}) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.3

Multiply ${\sqrt{2 (5 - b^{2})}}^{2}$ by $1$ .

$3 {\sqrt{2 (5 - b^{2})}}^{2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4

Rewrite ${\sqrt{2 (5 - b^{2})}}^{2}$ as $2 (5 - b^{2})$ .

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Step 3.1.2.1.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (5 - b^{2})}$ as ${(2 (5 - b^{2}))}^{\frac{1}{2}}$ .

$3 {({(2 (5 - b^{2}))}^{\frac{1}{2}})}^{2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4.2

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

$3 {(2 (5 - b^{2}))}^{\frac{1}{2} \cdot 2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4.3

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

$3 {(2 (5 - b^{2}))}^{\frac{2}{2}} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.1.4.4.1

Cancel the common factor.

$3 {(2 (5 - b^{2}))}^{\frac{2}{2}} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4.4.2

Rewrite the expression.

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.4.5

Simplify.

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$3 (2 (5 - b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.5

Apply the distributive property.

$3 (2 \cdot 5 + 2 (- b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.6

Multiply $2$ by $5$ .

$3 (10 + 2 (- b^{2})) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.7

Multiply $- 1$ by $2$ .

$3 (10 - 2 b^{2}) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.8

Apply the distributive property.

$3 \cdot 10 + 3 (- 2 b^{2}) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.9

Multiply $3$ by $10$ .

$30 + 3 (- 2 b^{2}) - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.1.10

Multiply $- 2$ by $3$ .

$30 - 6 b^{2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$30 - 6 b^{2} - b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.1.2.1.2

Subtract $b^{2}$ from $- 6 b^{2}$ .

$30 - 7 b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

$30 - 7 b^{2} = 9$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2

Solve for $b$ in $30 - 7 b^{2} = 9$ .

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Step 3.2.1

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

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Step 3.2.1.1

Subtract $30$ from both sides of the equation.

$- 7 b^{2} = 9 - 30$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.1.2

Subtract $30$ from $9$ .

$- 7 b^{2} = - 21$

$a = - \sqrt{2 (5 - b^{2})}$

$- 7 b^{2} = - 21$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.2

Divide each term in $- 7 b^{2} = - 21$ by $- 7$ and simplify.

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Step 3.2.2.1

Divide each term in $- 7 b^{2} = - 21$ by $- 7$ .

$\frac{- 7 b^{2}}{- 7} = \frac{- 21}{- 7}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $- 7$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{- 7 b^{2}}{- 7} = \frac{- 21}{- 7}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.2.2.1.2

Divide $b^{2}$ by $1$ .

$b^{2} = \frac{- 21}{- 7}$

$a = - \sqrt{2 (5 - b^{2})}$

$b^{2} = \frac{- 21}{- 7}$

$a = - \sqrt{2 (5 - b^{2})}$

$b^{2} = \frac{- 21}{- 7}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Divide $- 21$ by $- 7$ .

$b^{2} = 3$

$a = - \sqrt{2 (5 - b^{2})}$

$b^{2} = 3$

$a = - \sqrt{2 (5 - b^{2})}$

$b^{2} = 3$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$b = \pm \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$b = \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$b = - \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$b = \sqrt{3}, - \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

$b = \sqrt{3}, - \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

$b = \sqrt{3}, - \sqrt{3}$

$a = - \sqrt{2 (5 - b^{2})}$

Step 3.3

Replace all occurrences of $b$ with $\sqrt{3}$ in each equation.

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Step 3.3.1

Replace all occurrences of $b$ in $a = - \sqrt{2 (5 - b^{2})}$ with $\sqrt{3}$ .

$a = - \sqrt{2 (5 - {(\sqrt{3})}^{2})}$

$b = \sqrt{3}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $- \sqrt{2 (5 - {(\sqrt{3})}^{2})}$ .

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Step 3.3.2.1.1

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

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Step 3.3.2.1.1.1

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

$a = - \sqrt{2 (5 - {(3^{\frac{1}{2}})}^{2})}$

$b = \sqrt{3}$

Step 3.3.2.1.1.2

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

$a = - \sqrt{2 (5 - 3^{\frac{1}{2} \cdot 2})}$

$b = \sqrt{3}$

Step 3.3.2.1.1.3

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

$a = - \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = \sqrt{3}$

Step 3.3.2.1.1.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.4.1

Cancel the common factor.

$a = - \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = \sqrt{3}$

Step 3.3.2.1.1.4.2

Rewrite the expression.

$a = - \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

$a = - \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

Step 3.3.2.1.1.5

Evaluate the exponent.

$a = - \sqrt{2 (5 - 1 \cdot 3)}$

$b = \sqrt{3}$

$a = - \sqrt{2 (5 - 1 \cdot 3)}$

$b = \sqrt{3}$

Step 3.3.2.1.2

Simplify the expression.

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Step 3.3.2.1.2.1

Multiply $- 1$ by $3$ .

$a = - \sqrt{2 (5 - 3)}$

$b = \sqrt{3}$

Step 3.3.2.1.2.2

Subtract $3$ from $5$ .

$a = - \sqrt{2 \cdot 2}$

$b = \sqrt{3}$

Step 3.3.2.1.2.3

Multiply $2$ by $2$ .

$a = - \sqrt{4}$

$b = \sqrt{3}$

Step 3.3.2.1.2.4

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

$a = - \sqrt{2^{2}}$

$b = \sqrt{3}$

$a = - \sqrt{2^{2}}$

$b = \sqrt{3}$

Step 3.3.2.1.3

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

$a = - 1 \cdot 2$

$b = \sqrt{3}$

Step 3.3.2.1.4

Multiply $- 1$ by $2$ .

$a = - 2$

$b = \sqrt{3}$

$a = - 2$

$b = \sqrt{3}$

$a = - 2$

$b = \sqrt{3}$

$a = - 2$

$b = \sqrt{3}$

Step 3.4

Replace all occurrences of $b$ with $- \sqrt{3}$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $a = - \sqrt{2 (5 - b^{2})}$ with $- \sqrt{3}$ .

$a = - \sqrt{2 (5 - {(- \sqrt{3})}^{2})}$

$b = - \sqrt{3}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $- \sqrt{2 (5 - {(- \sqrt{3})}^{2})}$ .

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Step 3.4.2.1.1

Apply the product rule to $- \sqrt{3}$ .

$a = - \sqrt{2 (5 - ({(- 1)}^{2} {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 3.4.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 3.4.2.1.2.1

Move ${(- 1)}^{2}$ .

$a = - \sqrt{2 (5 + {(- 1)}^{2} \cdot (- 1 {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 3.4.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 3.4.2.1.2.2.1

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

$a = - \sqrt{2 (5 + {(- 1)}^{2} \cdot ((- 1) {\sqrt{3}}^{2}))}$

$b = - \sqrt{3}$

Step 3.4.2.1.2.2.2

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

$a = - \sqrt{2 (5 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

$a = - \sqrt{2 (5 + {(- 1)}^{2 + 1} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 3.4.2.1.2.3

Add $2$ and $1$ .

$a = - \sqrt{2 (5 + {(- 1)}^{3} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

$a = - \sqrt{2 (5 + {(- 1)}^{3} {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 3.4.2.1.3

Raise $- 1$ to the power of $3$ .

$a = - \sqrt{2 (5 - {\sqrt{3}}^{2})}$

$b = - \sqrt{3}$

Step 3.4.2.1.4

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

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Step 3.4.2.1.4.1

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

$a = - \sqrt{2 (5 - {(3^{\frac{1}{2}})}^{2})}$

$b = - \sqrt{3}$

Step 3.4.2.1.4.2

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

$a = - \sqrt{2 (5 - 3^{\frac{1}{2} \cdot 2})}$

$b = - \sqrt{3}$

Step 3.4.2.1.4.3

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

$a = - \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = - \sqrt{3}$

Step 3.4.2.1.4.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.4.4.1

Cancel the common factor.

$a = - \sqrt{2 (5 - 3^{\frac{2}{2}})}$

$b = - \sqrt{3}$

Step 3.4.2.1.4.4.2

Rewrite the expression.

$a = - \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

$a = - \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

Step 3.4.2.1.4.5

Evaluate the exponent.

$a = - \sqrt{2 (5 - 1 \cdot 3)}$

$b = - \sqrt{3}$

$a = - \sqrt{2 (5 - 1 \cdot 3)}$

$b = - \sqrt{3}$

Step 3.4.2.1.5

Simplify the expression.

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Step 3.4.2.1.5.1

Multiply $- 1$ by $3$ .

$a = - \sqrt{2 (5 - 3)}$

$b = - \sqrt{3}$

Step 3.4.2.1.5.2

Subtract $3$ from $5$ .

$a = - \sqrt{2 \cdot 2}$

$b = - \sqrt{3}$

Step 3.4.2.1.5.3

Multiply $2$ by $2$ .

$a = - \sqrt{4}$

$b = - \sqrt{3}$

Step 3.4.2.1.5.4

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

$a = - \sqrt{2^{2}}$

$b = - \sqrt{3}$

$a = - \sqrt{2^{2}}$

$b = - \sqrt{3}$

Step 3.4.2.1.6

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

$a = - 1 \cdot 2$

$b = - \sqrt{3}$

Step 3.4.2.1.7

Multiply $- 1$ by $2$ .

$a = - 2$

$b = - \sqrt{3}$

$a = - 2$

$b = - \sqrt{3}$

$a = - 2$

$b = - \sqrt{3}$

$a = - 2$

$b = - \sqrt{3}$

$a = - 2$

$b = - \sqrt{3}$

Step 4

List all of the solutions.

$a = 2, b = \sqrt{3}$

$a = 2, b = - \sqrt{3}$

$a = - 2, b = \sqrt{3}$

$a = - 2, b = - \sqrt{3}$