Trigonometry Examples

$2 \log (x) = \log (x^{2} + 2 x - 5)$

Step 1

Subtract $\log (x^{2} + 2 x - 5)$ from both sides of the equation.

$2 \log (x) - \log (x^{2} + 2 x - 5) = 0$

Step 2

Simplify $2 \log (x) - \log (x^{2} + 2 x - 5)$ .

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Step 2.1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) - \log (x^{2} + 2 x - 5) = 0$

Step 2.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{x^{2}}{x^{2} + 2 x - 5}) = 0$

Step 3

Set the denominator in $\frac{x^{2}}{x^{2} + 2 x - 5}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 2 x - 5 = 0$

Step 4

Solve for $x$ .

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Step 4.1

Use the quadratic formula to find the solutions.

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

Step 4.2

Substitute the values $a = 1$ , $b = 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 4.3.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.3.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 4.3.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 4.3.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 4.3.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 4.4

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

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 4.4.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.4.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 4.4.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 4.4.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{6}$

Step 4.5

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

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 4.5.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.5.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 4.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{6}$

Step 4.6

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{6}, - 1 - \sqrt{6}$

Step 5

Set the argument in $\log (\frac{x^{2}}{x^{2} + 2 x - 5})$ less than or equal to $0$ to find where the expression is undefined.

$\frac{x^{2}}{x^{2} + 2 x - 5} \leq 0$

Step 6

Solve for $x$ .

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Step 6.1

Find all the values where the expression switches from negative to positive by setting each factor equal to $0$ and solving.

$x^{2} = 0$

$x^{2} + 2 x - 5 = 0$

Step 6.2

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

$x = \pm \sqrt{0}$

Step 6.3

Simplify $\pm \sqrt{0}$ .

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Step 6.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.3.2

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

$x = \pm 0$

Step 6.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 6.4

Use the quadratic formula to find the solutions.

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

Step 6.5

Substitute the values $a = 1$ , $b = 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 6.6

Simplify.

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Step 6.6.1

Simplify the numerator.

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Step 6.6.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.6.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.6.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.6.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.6.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.6.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.6.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.6.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.6.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.6.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.7

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

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Step 6.7.1

Simplify the numerator.

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Step 6.7.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.7.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.7.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.7.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.7.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.7.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.7.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.7.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.7.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.7.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.7.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{6}$

Step 6.8

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

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Step 6.8.1

Simplify the numerator.

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Step 6.8.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.8.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.8.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.8.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.8.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.8.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.8.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.8.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.8.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.8.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.8.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{6}$

Step 6.9

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{6}, - 1 - \sqrt{6}$

Step 6.10

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

$x = 0$

$x = - 1 + \sqrt{6}, - 1 - \sqrt{6}$

Step 6.11

Consolidate the solutions.

$x = 0, - 1 + \sqrt{6}, - 1 - \sqrt{6}$

Step 6.12

Find the domain of $\frac{x^{2}}{x^{2} + 2 x - 5}$ .

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Step 6.12.1

Set the denominator in $\frac{x^{2}}{x^{2} + 2 x - 5}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 2 x - 5 = 0$

Step 6.12.2

Solve for $x$ .

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Step 6.12.2.1

Use the quadratic formula to find the solutions.

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

Step 6.12.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 6.12.2.3

Simplify.

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Step 6.12.2.3.1

Simplify the numerator.

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Step 6.12.2.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.12.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.3.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.12.2.3.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.12.2.3.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.12.2.3.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.12.2.3.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.12.2.3.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.12.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.12.2.3.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.12.2.4

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

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Step 6.12.2.4.1

Simplify the numerator.

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Step 6.12.2.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.12.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.4.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.12.2.4.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.12.2.4.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.12.2.4.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.12.2.4.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.12.2.4.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.12.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.12.2.4.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.12.2.4.4

Change the $\pm$ to $+$ .

$x = - 1 + \sqrt{6}$

Step 6.12.2.5

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

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Step 6.12.2.5.1

Simplify the numerator.

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Step 6.12.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 6.12.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 5}}{2 \cdot 1}$

Step 6.12.2.5.1.2.2

Multiply $- 4$ by $- 5$ .

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

Step 6.12.2.5.1.3

Add $4$ and $20$ .

$x = \frac{- 2 \pm \sqrt{24}}{2 \cdot 1}$

Step 6.12.2.5.1.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 6.12.2.5.1.4.1

Factor $4$ out of $24$ .

$x = \frac{- 2 \pm \sqrt{4 (6)}}{2 \cdot 1}$

Step 6.12.2.5.1.4.2

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

$x = \frac{- 2 \pm \sqrt{2^{2} \cdot 6}}{2 \cdot 1}$

Step 6.12.2.5.1.5

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{6}}{2 \cdot 1}$

Step 6.12.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

Step 6.12.2.5.3

Simplify $\frac{- 2 \pm 2 \sqrt{6}}{2}$ .

$x = - 1 \pm \sqrt{6}$

Step 6.12.2.5.4

Change the $\pm$ to $-$ .

$x = - 1 - \sqrt{6}$

Step 6.12.2.6

The final answer is the combination of both solutions.

$x = - 1 + \sqrt{6}, - 1 - \sqrt{6}$

Step 6.12.3

The domain is all values of $x$ that make the expression defined.

$(- \infty, - 1 - \sqrt{6}) \cup (- 1 - \sqrt{6}, - 1 + \sqrt{6}) \cup (- 1 + \sqrt{6}, \infty)$

Step 6.13

Use each root to create test intervals.

$x < - 1 - \sqrt{6}$

$- 1 - \sqrt{6} < x < 0$

$0 < x < - 1 + \sqrt{6}$

$x > - 1 + \sqrt{6}$

Step 6.14

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

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Step 6.14.1

Test a value on the interval $x < - 1 - \sqrt{6}$ to see if it makes the inequality true.

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Step 6.14.1.1

Choose a value on the interval $x < - 1 - \sqrt{6}$ and see if this value makes the original inequality true.

$x = - 6$

Step 6.14.1.2

Replace $x$ with $- 6$ in the original inequality.

$\frac{{(- 6)}^{2}}{{(- 6)}^{2} + 2 (- 6) - 5} \leq 0$

Step 6.14.1.3

The left side $1.89473684$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 6.14.2

Test a value on the interval $- 1 - \sqrt{6} < x < 0$ to see if it makes the inequality true.

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Step 6.14.2.1

Choose a value on the interval $- 1 - \sqrt{6} < x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Step 6.14.2.2

Replace $x$ with $- 2$ in the original inequality.

$\frac{{(- 2)}^{2}}{{(- 2)}^{2} + 2 (- 2) - 5} \leq 0$

Step 6.14.2.3

The left side $- 0.8$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.14.3

Test a value on the interval $0 < x < - 1 + \sqrt{6}$ to see if it makes the inequality true.

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Step 6.14.3.1

Choose a value on the interval $0 < x < - 1 + \sqrt{6}$ and see if this value makes the original inequality true.

$x = 1$

Step 6.14.3.2

Replace $x$ with $1$ in the original inequality.

$\frac{{(1)}^{2}}{{(1)}^{2} + 2 (1) - 5} \leq 0$

Step 6.14.3.3

The left side $- 0.5$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 6.14.4

Test a value on the interval $x > - 1 + \sqrt{6}$ to see if it makes the inequality true.

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Step 6.14.4.1

Choose a value on the interval $x > - 1 + \sqrt{6}$ and see if this value makes the original inequality true.

$x = 4$

Step 6.14.4.2

Replace $x$ with $4$ in the original inequality.

$\frac{{(4)}^{2}}{{(4)}^{2} + 2 (4) - 5} \leq 0$

Step 6.14.4.3

The left side $0.84210526$ is greater than the right side $0$ , which means that the given statement is false.

False

Step 6.14.5

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 1 - \sqrt{6}$ False

$- 1 - \sqrt{6} < x < 0$ True

$0 < x < - 1 + \sqrt{6}$ True

$x > - 1 + \sqrt{6}$ False

$x < - 1 - \sqrt{6}$ False

$- 1 - \sqrt{6} < x < 0$ True

$0 < x < - 1 + \sqrt{6}$ True

$x > - 1 + \sqrt{6}$ False

Step 6.15

The solution consists of all of the true intervals.

$- 1 - \sqrt{6} < x \leq 0$ or $0 \leq x < - 1 + \sqrt{6}$

Step 6.16

Combine the intervals.

$- 1 - \sqrt{6} < x < - 1 + \sqrt{6}$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$- 1 - \sqrt{6} \leq x \leq - 1 + \sqrt{6}$

$[- 1 - \sqrt{6}, - 1 + \sqrt{6}]$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$- 1 - \sqrt{6} \leq x \leq - 1 + \sqrt{6} [- 1 - \sqrt{6}, - 1 + \sqrt{6}]$

Interval Notation:

$[- 1 - \sqrt{6}, - 1 + \sqrt{6})$

Step 9