Algebra Examples

$\log_{5} (4 x^{2} - 6) - \log_{5} (4 x + 1) = 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{5} (\frac{4 x^{2} - 6}{4 x + 1}) = 1$

Step 1.2

Factor $2$ out of $4 x^{2} - 6$ .

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Step 1.2.1

Factor $2$ out of $4 x^{2}$ .

$\log_{5} (\frac{2 (2 x^{2}) - 6}{4 x + 1}) = 1$

Step 1.2.2

Factor $2$ out of $- 6$ .

$\log_{5} (\frac{2 (2 x^{2}) + 2 (- 3)}{4 x + 1}) = 1$

Step 1.2.3

Factor $2$ out of $2 (2 x^{2}) + 2 (- 3)$ .

$\log_{5} (\frac{2 (2 x^{2} - 3)}{4 x + 1}) = 1$

Step 2

Rewrite $\log_{5} (\frac{2 (2 x^{2} - 3)}{4 x + 1}) = 1$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$5^{1} = \frac{2 (2 x^{2} - 3)}{4 x + 1}$

Step 3

Cross multiply to remove the fraction.

$2 (2 x^{2} - 3) = 5 (4 x + 1)$

Step 4

Simplify $5 (4 x + 1)$ .

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Step 4.1

Apply the distributive property.

$2 (2 x^{2} - 3) = 5 (4 x) + 5 \cdot 1$

Step 4.2

Multiply.

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Step 4.2.1

Multiply $4$ by $5$ .

$2 (2 x^{2} - 3) = 20 x + 5 \cdot 1$

Step 4.2.2

Multiply $5$ by $1$ .

$2 (2 x^{2} - 3) = 20 x + 5$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $20 x$ from both sides of the equation.

$2 (2 x^{2} - 3) - 20 x = 5$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$2 (2 x^{2}) + 2 \cdot - 3 - 20 x = 5$

Step 5.2.2

Multiply $2$ by $2$ .

$4 x^{2} + 2 \cdot - 3 - 20 x = 5$

Step 5.2.3

Multiply $2$ by $- 3$ .

$4 x^{2} - 6 - 20 x = 5$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $2$ out of $4 x^{2} - 6 - 20 x$ .

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Step 6.1.1

Factor $2$ out of $4 x^{2}$ .

$2 (2 x^{2}) - 6 - 20 x = 5$

Step 6.1.2

Factor $2$ out of $- 6$ .

$2 (2 x^{2}) + 2 (- 3) - 20 x = 5$

Step 6.1.3

Factor $2$ out of $- 20 x$ .

$2 (2 x^{2}) + 2 (- 3) + 2 (- 10 x) = 5$

Step 6.1.4

Factor $2$ out of $2 (2 x^{2}) + 2 (- 3)$ .

$2 (2 x^{2} - 3) + 2 (- 10 x) = 5$

Step 6.1.5

Factor $2$ out of $2 (2 x^{2} - 3) + 2 (- 10 x)$ .

$2 (2 x^{2} - 3 - 10 x) = 5$

Step 6.2

Reorder terms.

$2 (2 x^{2} - 10 x - 3) = 5$

Step 7

Divide each term in $2 (2 x^{2} - 10 x - 3) = 5$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 (2 x^{2} - 10 x - 3) = 5$ by $2$ .

$\frac{2 (2 x^{2} - 10 x - 3)}{2} = \frac{5}{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 (2 x^{2} - 10 x - 3)}{2} = \frac{5}{2}$

Step 7.2.1.2

Divide $2 x^{2} - 10 x - 3$ by $1$ .

$2 x^{2} - 10 x - 3 = \frac{5}{2}$

Step 8

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

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Step 8.1

Subtract $\frac{5}{2}$ from both sides of the equation.

$2 x^{2} - 10 x - 3 - \frac{5}{2} = 0$

Step 8.2

Simplify $2 x^{2} - 10 x - 3 - \frac{5}{2}$ .

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Step 8.2.1

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x^{2} - 10 x - 3 \cdot \frac{2}{2} - \frac{5}{2} = 0$

Step 8.2.2

Combine $- 3$ and $\frac{2}{2}$ .

$2 x^{2} - 10 x + \frac{- 3 \cdot 2}{2} - \frac{5}{2} = 0$

Step 8.2.3

Combine the numerators over the common denominator.

$2 x^{2} - 10 x + \frac{- 3 \cdot 2 - 5}{2} = 0$

Step 8.2.4

Simplify the numerator.

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Step 8.2.4.1

Multiply $- 3$ by $2$ .

$2 x^{2} - 10 x + \frac{- 6 - 5}{2} = 0$

Step 8.2.4.2

Subtract $5$ from $- 6$ .

$2 x^{2} - 10 x + \frac{- 11}{2} = 0$

Step 8.2.5

Move the negative in front of the fraction.

$2 x^{2} - 10 x - \frac{11}{2} = 0$

Step 9

Multiply through by the least common denominator $2$ , then simplify.

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Step 9.1

Apply the distributive property.

$2 (2 x^{2}) + 2 (- 10 x) + 2 (- \frac{11}{2}) = 0$

Step 9.2

Simplify.

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Step 9.2.1

Multiply $2$ by $2$ .

$4 x^{2} + 2 (- 10 x) + 2 (- \frac{11}{2}) = 0$

Step 9.2.2

Multiply $- 10$ by $2$ .

$4 x^{2} - 20 x + 2 (- \frac{11}{2}) = 0$

Step 9.2.3

Cancel the common factor of $2$ .

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Step 9.2.3.1

Move the leading negative in $- \frac{11}{2}$ into the numerator.

$4 x^{2} - 20 x + 2 (\frac{- 11}{2}) = 0$

Step 9.2.3.2

Cancel the common factor.

$4 x^{2} - 20 x + 2 (\frac{- 11}{2}) = 0$

Step 9.2.3.3

Rewrite the expression.

$4 x^{2} - 20 x - 11 = 0$

Step 10

Use the quadratic formula to find the solutions.

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

Step 11

Substitute the values $a = 4$ , $b = - 20$ , and $c = - 11$ into the quadratic formula and solve for $x$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (4 \cdot - 11)}}{2 \cdot 4}$

Step 12

Simplify.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

Raise $- 20$ to the power of $2$ .

$x = \frac{20 \pm \sqrt{400 - 4 \cdot 4 \cdot - 11}}{2 \cdot 4}$

Step 12.1.2

Multiply $- 4 \cdot 4 \cdot - 11$ .

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Step 12.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{20 \pm \sqrt{400 - 16 \cdot - 11}}{2 \cdot 4}$

Step 12.1.2.2

Multiply $- 16$ by $- 11$ .

$x = \frac{20 \pm \sqrt{400 + 176}}{2 \cdot 4}$

Step 12.1.3

Add $400$ and $176$ .

$x = \frac{20 \pm \sqrt{576}}{2 \cdot 4}$

Step 12.1.4

Rewrite $576$ as $24^{2}$ .

$x = \frac{20 \pm \sqrt{24^{2}}}{2 \cdot 4}$

Step 12.1.5

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

$x = \frac{20 \pm 24}{2 \cdot 4}$

Step 12.2

Multiply $2$ by $4$ .

$x = \frac{20 \pm 24}{8}$

Step 12.3

Simplify $\frac{20 \pm 24}{8}$ .

$x = \frac{5 \pm 6}{2}$

Step 13

The final answer is the combination of both solutions.

$x = \frac{11}{2}, - \frac{1}{2}$

Step 14

Exclude the solutions that do not make $\log_{5} (4 x^{2} - 6) - \log_{5} (4 x + 1) = 1$ true.

$x = \frac{11}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{2}$

Decimal Form:

$x = 5.5$

Mixed Number Form:

$x = 5 \frac{1}{2}$