Algebra Examples

$\frac{1}{x} + \frac{1}{x + 4} = \frac{1}{5}$

Step 1

Subtract $\frac{1}{5}$ from both sides of the equation.

$\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Step 2

Simplify $\frac{1}{x} + \frac{1}{x + 4} - \frac{1}{5}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Multiply $\frac{1}{x}$ by $\frac{(x + 4) \cdot 5}{(x + 4) \cdot 5}$ .

$\frac{1}{x} \cdot \frac{(x + 4) \cdot 5}{(x + 4) \cdot 5} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Step 2.1.2

Multiply $\frac{1}{x}$ by $\frac{(x + 4) \cdot 5}{(x + 4) \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{1}{x + 4} - \frac{1}{5} = 0$

Step 2.1.3

Multiply $\frac{1}{x + 4}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{1}{x + 4} \cdot \frac{x \cdot 5}{x \cdot 5} - \frac{1}{5} = 0$

Step 2.1.4

Multiply $\frac{1}{x + 4}$ by $\frac{x \cdot 5}{x \cdot 5}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{1}{5} = 0$

Step 2.1.5

Multiply $\frac{1}{5}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - (\frac{1}{5} \cdot \frac{x (x + 4)}{x (x + 4)}) = 0$

Step 2.1.6

Multiply $\frac{1}{5}$ by $\frac{x (x + 4)}{x (x + 4)}$ .

$\frac{(x + 4) \cdot 5}{x ((x + 4) \cdot 5)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Step 2.1.7

Reorder the factors of $x ((x + 4) \cdot 5)$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{(x + 4) (x \cdot 5)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Step 2.1.8

Reorder the factors of $(x + 4) (x \cdot 5)$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{5 x (x + 4)} - \frac{x (x + 4)}{5 (x (x + 4))} = 0$

Step 2.1.9

Reorder the factors of $5 (x (x + 4))$ .

$\frac{(x + 4) \cdot 5}{5 x (x + 4)} + \frac{x \cdot 5}{5 x (x + 4)} - \frac{x (x + 4)}{5 x (x + 4)} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{(x + 4) \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

Apply the distributive property.

$\frac{x \cdot 5 + 4 \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Step 2.3.2

Move $5$ to the left of $x$ .

$\frac{5 \cdot x + 4 \cdot 5 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Step 2.3.3

Multiply $4$ by $5$ .

$\frac{5 \cdot x + 20 + x \cdot 5 - x (x + 4)}{5 x (x + 4)} = 0$

Step 2.3.4

Move $5$ to the left of $x$ .

$\frac{5 x + 20 + 5 \cdot x - x (x + 4)}{5 x (x + 4)} = 0$

Step 2.3.5

Apply the distributive property.

$\frac{5 x + 20 + 5 x - x \cdot x - x \cdot 4}{5 x (x + 4)} = 0$

Step 2.3.6

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

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Step 2.3.6.1

Move $x$ .

$\frac{5 x + 20 + 5 x - (x \cdot x) - x \cdot 4}{5 x (x + 4)} = 0$

Step 2.3.6.2

Multiply $x$ by $x$ .

$\frac{5 x + 20 + 5 x - x^{2} - x \cdot 4}{5 x (x + 4)} = 0$

Step 2.3.7

Multiply $4$ by $- 1$ .

$\frac{5 x + 20 + 5 x - x^{2} - 4 x}{5 x (x + 4)} = 0$

Step 2.4

Add $5 x$ and $5 x$ .

$\frac{10 x + 20 - x^{2} - 4 x}{5 x (x + 4)} = 0$

Step 2.5

Subtract $4 x$ from $10 x$ .

$\frac{6 x + 20 - x^{2}}{5 x (x + 4)} = 0$

Step 2.6

Reorder terms.

$\frac{- x^{2} + 6 x + 20}{5 x (x + 4)} = 0$

Step 2.7

Factor $- 1$ out of $- x^{2}$ .

$\frac{- (x^{2}) + 6 x + 20}{5 x (x + 4)} = 0$

Step 2.8

Factor $- 1$ out of $6 x$ .

$\frac{- (x^{2}) - (- 6 x) + 20}{5 x (x + 4)} = 0$

Step 2.9

Factor $- 1$ out of $- (x^{2}) - (- 6 x)$ .

$\frac{- (x^{2} - 6 x) + 20}{5 x (x + 4)} = 0$

Step 2.10

Rewrite $20$ as $- 1 (- 20)$ .

$\frac{- (x^{2} - 6 x) - 1 (- 20)}{5 x (x + 4)} = 0$

Step 2.11

Factor $- 1$ out of $- (x^{2} - 6 x) - 1 (- 20)$ .

$\frac{- (x^{2} - 6 x - 20)}{5 x (x + 4)} = 0$

Step 2.12

Rewrite $- (x^{2} - 6 x - 20)$ as $- 1 (x^{2} - 6 x - 20)$ .

$\frac{- 1 (x^{2} - 6 x - 20)}{5 x (x + 4)} = 0$

Step 2.13

Move the negative in front of the fraction.

$- \frac{x^{2} - 6 x - 20}{5 x (x + 4)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 6 x - 20 = 0$

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 $a = 1$ , $b = - 6$ , and $c = - 20$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (1 \cdot - 20)}}{2 \cdot 1}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Step 4.3.1.3

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 4.3.1.4.1

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 4.3.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 4.3.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Step 4.3.3

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Step 4.4.1.3

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 4.4.1.4.1

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 4.4.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Step 4.4.3

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = 3 + \sqrt{29}$

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 $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 1 \cdot - 20}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot - 20$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot - 20}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $- 20$ .

$x = \frac{6 \pm \sqrt{36 + 80}}{2 \cdot 1}$

Step 4.5.1.3

Add $36$ and $80$ .

$x = \frac{6 \pm \sqrt{116}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $116$ as $2^{2} \cdot 29$ .

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Step 4.5.1.4.1

Factor $4$ out of $116$ .

$x = \frac{6 \pm \sqrt{4 (29)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 29}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{29}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{6 \pm 2 \sqrt{29}}{2}$

Step 4.5.3

Simplify $\frac{6 \pm 2 \sqrt{29}}{2}$ .

$x = 3 \pm \sqrt{29}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = 3 - \sqrt{29}$

Step 4.6

The final answer is the combination of both solutions.

$x = 3 + \sqrt{29}, 3 - \sqrt{29}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 3 + \sqrt{29}, 3 - \sqrt{29}$

Decimal Form:

$x = 8.38516480 \dots, - 2.38516480 \dots$