Finite Math Examples

$\frac{6}{x - 6} - \frac{7}{x + 6} = \frac{6}{2^{2} - 36}$

Step 1

Subtract $\frac{6}{2^{2} - 36}$ from both sides of the equation.

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{2^{2} - 36} = 0$

Step 2

Simplify $\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{2^{2} - 36}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify the denominator.

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Step 2.1.1.1

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

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{4 - 36} = 0$

Step 2.1.1.2

Subtract $36$ from $4$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{- 32} = 0$

Step 2.1.2

Cancel the common factor of $6$ and $- 32$ .

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Step 2.1.2.1

Factor $2$ out of $6$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{2 (3)}{- 32} = 0$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Factor $2$ out of $- 32$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{2 \cdot 3}{2 \cdot - 16} = 0$

Step 2.1.2.2.2

Cancel the common factor.

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{2 \cdot 3}{2 \cdot - 16} = 0$

Step 2.1.2.2.3

Rewrite the expression.

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{3}{- 16} = 0$

Step 2.1.3

Move the negative in front of the fraction.

$\frac{6}{x - 6} - \frac{7}{x + 6} - - \frac{3}{16} = 0$

Step 2.1.4

Multiply $- - \frac{3}{16}$ .

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Step 2.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} + 1 (\frac{3}{16}) = 0$

Step 2.1.4.2

Multiply $\frac{3}{16}$ by $1$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} + \frac{3}{16} = 0$

Step 2.2

To write $\frac{6}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{6}{x - 6} \cdot \frac{x + 6}{x + 6} - \frac{7}{x + 6} + \frac{3}{16} = 0$

Step 2.3

To write $- \frac{7}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{6}{x - 6} \cdot \frac{x + 6}{x + 6} - \frac{7}{x + 6} \cdot \frac{x - 6}{x - 6} + \frac{3}{16} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 6) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{6}{x - 6}$ by $\frac{x + 6}{x + 6}$ .

$\frac{6 (x + 6)}{(x - 6) (x + 6)} - \frac{7}{x + 6} \cdot \frac{x - 6}{x - 6} + \frac{3}{16} = 0$

Step 2.4.2

Multiply $\frac{7}{x + 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{6 (x + 6)}{(x - 6) (x + 6)} - \frac{7 (x - 6)}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.4.3

Reorder the factors of $(x - 6) (x + 6)$ .

$\frac{6 (x + 6)}{(x + 6) (x - 6)} - \frac{7 (x - 6)}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{6 (x + 6) - 7 (x - 6)}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Apply the distributive property.

$\frac{6 x + 6 \cdot 6 - 7 (x - 6)}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6.2

Multiply $6$ by $6$ .

$\frac{6 x + 36 - 7 (x - 6)}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6.3

Apply the distributive property.

$\frac{6 x + 36 - 7 x - 7 \cdot - 6}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6.4

Multiply $- 7$ by $- 6$ .

$\frac{6 x + 36 - 7 x + 42}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6.5

Subtract $7 x$ from $6 x$ .

$\frac{- x + 36 + 42}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.6.6

Add $36$ and $42$ .

$\frac{- x + 78}{(x + 6) (x - 6)} + \frac{3}{16} = 0$

Step 2.7

To write $\frac{- x + 78}{(x + 6) (x - 6)}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$\frac{- x + 78}{(x + 6) (x - 6)} \cdot \frac{16}{16} + \frac{3}{16} = 0$

Step 2.8

To write $\frac{3}{16}$ as a fraction with a common denominator, multiply by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{- x + 78}{(x + 6) (x - 6)} \cdot \frac{16}{16} + \frac{3}{16} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)} = 0$

Step 2.9

Write each expression with a common denominator of $(x + 6) (x - 6) \cdot 16$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.9.1

Multiply $\frac{- x + 78}{(x + 6) (x - 6)}$ by $\frac{16}{16}$ .

$\frac{(- x + 78) \cdot 16}{(x + 6) (x - 6) \cdot 16} + \frac{3}{16} \cdot \frac{(x + 6) (x - 6)}{(x + 6) (x - 6)} = 0$

Step 2.9.2

Multiply $\frac{3}{16}$ by $\frac{(x + 6) (x - 6)}{(x + 6) (x - 6)}$ .

$\frac{(- x + 78) \cdot 16}{(x + 6) (x - 6) \cdot 16} + \frac{3 ((x + 6) (x - 6))}{16 ((x + 6) (x - 6))} = 0$

Step 2.9.3

Reorder the factors of $(x + 6) (x - 6) \cdot 16$ .

$\frac{(- x + 78) \cdot 16}{16 (x + 6) (x - 6)} + \frac{3 ((x + 6) (x - 6))}{16 ((x + 6) (x - 6))} = 0$

Step 2.9.4

Reorder the factors of $16 ((x + 6) (x - 6))$ .

$\frac{(- x + 78) \cdot 16}{16 (x + 6) (x - 6)} + \frac{3 ((x + 6) (x - 6))}{16 (x + 6) (x - 6)} = 0$

Step 2.10

Combine the numerators over the common denominator.

$\frac{(- x + 78) \cdot 16 + 3 ((x + 6) (x - 6))}{16 (x + 6) (x - 6)} = 0$

Step 2.11

Simplify the numerator.

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Step 2.11.1

Apply the distributive property.

$\frac{- x \cdot 16 + 78 \cdot 16 + 3 (x + 6) (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.2

Multiply $16$ by $- 1$ .

$\frac{- 16 x + 78 \cdot 16 + 3 (x + 6) (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.3

Multiply $78$ by $16$ .

$\frac{- 16 x + 1248 + 3 (x + 6) (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.4

Apply the distributive property.

$\frac{- 16 x + 1248 + (3 x + 3 \cdot 6) (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.5

Multiply $3$ by $6$ .

$\frac{- 16 x + 1248 + (3 x + 18) (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.6

Expand $(3 x + 18) (x - 6)$ using the FOIL Method.

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Step 2.11.6.1

Apply the distributive property.

$\frac{- 16 x + 1248 + 3 x (x - 6) + 18 (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.6.2

Apply the distributive property.

$\frac{- 16 x + 1248 + 3 x \cdot x + 3 x \cdot - 6 + 18 (x - 6)}{16 (x + 6) (x - 6)} = 0$

Step 2.11.6.3

Apply the distributive property.

$\frac{- 16 x + 1248 + 3 x \cdot x + 3 x \cdot - 6 + 18 x + 18 \cdot - 6}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7

Simplify and combine like terms.

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Step 2.11.7.1

Simplify each term.

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Step 2.11.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.11.7.1.1.1

Move $x$ .

$\frac{- 16 x + 1248 + 3 (x \cdot x) + 3 x \cdot - 6 + 18 x + 18 \cdot - 6}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7.1.1.2

Multiply $x$ by $x$ .

$\frac{- 16 x + 1248 + 3 x^{2} + 3 x \cdot - 6 + 18 x + 18 \cdot - 6}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7.1.2

Multiply $- 6$ by $3$ .

$\frac{- 16 x + 1248 + 3 x^{2} - 18 x + 18 x + 18 \cdot - 6}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7.1.3

Multiply $18$ by $- 6$ .

$\frac{- 16 x + 1248 + 3 x^{2} - 18 x + 18 x - 108}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7.2

Add $- 18 x$ and $18 x$ .

$\frac{- 16 x + 1248 + 3 x^{2} + 0 - 108}{16 (x + 6) (x - 6)} = 0$

Step 2.11.7.3

Add $3 x^{2}$ and $0$ .

$\frac{- 16 x + 1248 + 3 x^{2} - 108}{16 (x + 6) (x - 6)} = 0$

Step 2.11.8

Subtract $108$ from $1248$ .

$\frac{- 16 x + 3 x^{2} + 1140}{16 (x + 6) (x - 6)} = 0$

Step 2.11.9

Reorder terms.

$\frac{3 x^{2} - 16 x + 1140}{16 (x + 6) (x - 6)} = 0$

Step 3

Set the numerator equal to zero.

$3 x^{2} - 16 x + 1140 = 0$

Step 4

Solve the equation 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 $a = 3$ , $b = - 16$ , and $c = 1140$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (3 \cdot 1140)}}{2 \cdot 3}$

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 $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot 1140}}{2 \cdot 3}$

Step 4.3.1.2

Multiply $- 4 \cdot 3 \cdot 1140$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot 1140}}{2 \cdot 3}$

Step 4.3.1.2.2

Multiply $- 12$ by $1140$ .

$x = \frac{16 \pm \sqrt{256 - 13680}}{2 \cdot 3}$

Step 4.3.1.3

Subtract $13680$ from $256$ .

$x = \frac{16 \pm \sqrt{- 13424}}{2 \cdot 3}$

Step 4.3.1.4

Rewrite $- 13424$ as $- 1 (13424)$ .

$x = \frac{16 \pm \sqrt{- 1 \cdot 13424}}{2 \cdot 3}$

Step 4.3.1.5

Rewrite $\sqrt{- 1 (13424)}$ as $\sqrt{- 1} \cdot \sqrt{13424}$ .

$x = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.3.1.6

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

$x = \frac{16 \pm i \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.3.1.7

Rewrite $13424$ as $4^{2} \cdot 839$ .

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Step 4.3.1.7.1

Factor $16$ out of $13424$ .

$x = \frac{16 \pm i \cdot \sqrt{16 (839)}}{2 \cdot 3}$

Step 4.3.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm i \cdot \sqrt{4^{2} \cdot 839}}{2 \cdot 3}$

Step 4.3.1.8

Pull terms out from under the radical.

$x = \frac{16 \pm i \cdot (4 \sqrt{839})}{2 \cdot 3}$

Step 4.3.1.9

Move $4$ to the left of $i$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{2 \cdot 3}$

Step 4.3.2

Multiply $2$ by $3$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{6}$

Step 4.3.3

Simplify $\frac{16 \pm 4 i \sqrt{839}}{6}$ .

$x = \frac{8 \pm 2 i \sqrt{839}}{3}$

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 $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot 1140}}{2 \cdot 3}$

Step 4.4.1.2

Multiply $- 4 \cdot 3 \cdot 1140$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot 1140}}{2 \cdot 3}$

Step 4.4.1.2.2

Multiply $- 12$ by $1140$ .

$x = \frac{16 \pm \sqrt{256 - 13680}}{2 \cdot 3}$

Step 4.4.1.3

Subtract $13680$ from $256$ .

$x = \frac{16 \pm \sqrt{- 13424}}{2 \cdot 3}$

Step 4.4.1.4

Rewrite $- 13424$ as $- 1 (13424)$ .

$x = \frac{16 \pm \sqrt{- 1 \cdot 13424}}{2 \cdot 3}$

Step 4.4.1.5

Rewrite $\sqrt{- 1 (13424)}$ as $\sqrt{- 1} \cdot \sqrt{13424}$ .

$x = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.4.1.6

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

$x = \frac{16 \pm i \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.4.1.7

Rewrite $13424$ as $4^{2} \cdot 839$ .

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Step 4.4.1.7.1

Factor $16$ out of $13424$ .

$x = \frac{16 \pm i \cdot \sqrt{16 (839)}}{2 \cdot 3}$

Step 4.4.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm i \cdot \sqrt{4^{2} \cdot 839}}{2 \cdot 3}$

Step 4.4.1.8

Pull terms out from under the radical.

$x = \frac{16 \pm i \cdot (4 \sqrt{839})}{2 \cdot 3}$

Step 4.4.1.9

Move $4$ to the left of $i$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{6}$

Step 4.4.3

Simplify $\frac{16 \pm 4 i \sqrt{839}}{6}$ .

$x = \frac{8 \pm 2 i \sqrt{839}}{3}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = \frac{8 + 2 i \sqrt{839}}{3}$

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 $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 3 \cdot 1140}}{2 \cdot 3}$

Step 4.5.1.2

Multiply $- 4 \cdot 3 \cdot 1140$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{16 \pm \sqrt{256 - 12 \cdot 1140}}{2 \cdot 3}$

Step 4.5.1.2.2

Multiply $- 12$ by $1140$ .

$x = \frac{16 \pm \sqrt{256 - 13680}}{2 \cdot 3}$

Step 4.5.1.3

Subtract $13680$ from $256$ .

$x = \frac{16 \pm \sqrt{- 13424}}{2 \cdot 3}$

Step 4.5.1.4

Rewrite $- 13424$ as $- 1 (13424)$ .

$x = \frac{16 \pm \sqrt{- 1 \cdot 13424}}{2 \cdot 3}$

Step 4.5.1.5

Rewrite $\sqrt{- 1 (13424)}$ as $\sqrt{- 1} \cdot \sqrt{13424}$ .

$x = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.5.1.6

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

$x = \frac{16 \pm i \cdot \sqrt{13424}}{2 \cdot 3}$

Step 4.5.1.7

Rewrite $13424$ as $4^{2} \cdot 839$ .

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Step 4.5.1.7.1

Factor $16$ out of $13424$ .

$x = \frac{16 \pm i \cdot \sqrt{16 (839)}}{2 \cdot 3}$

Step 4.5.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{16 \pm i \cdot \sqrt{4^{2} \cdot 839}}{2 \cdot 3}$

Step 4.5.1.8

Pull terms out from under the radical.

$x = \frac{16 \pm i \cdot (4 \sqrt{839})}{2 \cdot 3}$

Step 4.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{2 \cdot 3}$

Step 4.5.2

Multiply $2$ by $3$ .

$x = \frac{16 \pm 4 i \sqrt{839}}{6}$

Step 4.5.3

Simplify $\frac{16 \pm 4 i \sqrt{839}}{6}$ .

$x = \frac{8 \pm 2 i \sqrt{839}}{3}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = \frac{8 - 2 i \sqrt{839}}{3}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{8 + 2 i \sqrt{839}}{3}, \frac{8 - 2 i \sqrt{839}}{3}$