Algebra Examples

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

Step 1

Subtract $3$ from both sides of the equation.

$\frac{x}{x^{2} - 4} + \frac{1}{x + 2} - 3 = 0$

Step 2

Simplify $\frac{x}{x^{2} - 4} + \frac{1}{x + 2} - 3$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{x}{x^{2} - 2^{2}} + \frac{1}{x + 2} - 3 = 0$

Step 2.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$\frac{x}{(x + 2) (x - 2)} + \frac{1}{x + 2} - 3 = 0$

Step 2.2

To write $\frac{1}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{x}{(x + 2) (x - 2)} + \frac{1}{x + 2} \cdot \frac{x - 2}{x - 2} - 3 = 0$

Step 2.3

Multiply $\frac{1}{x + 2}$ by $\frac{x - 2}{x - 2}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Add $x$ and $x$ .

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

Step 2.6

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

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Step 2.6.1

Factor $2$ out of $2 x$ .

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

Step 2.6.2

Factor $2$ out of $- 2$ .

$\frac{2 (x) + 2 (- 1)}{(x + 2) (x - 2)} - 3 = 0$

Step 2.6.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$\frac{2 (x - 1)}{(x + 2) (x - 2)} - 3 = 0$

Step 2.7

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

$\frac{2 (x - 1)}{(x + 2) (x - 2)} - 3 \cdot \frac{(x + 2) (x - 2)}{(x + 2) (x - 2)} = 0$

Step 2.8

Combine $- 3$ and $\frac{(x + 2) (x - 2)}{(x + 2) (x - 2)}$ .

$\frac{2 (x - 1)}{(x + 2) (x - 2)} + \frac{- 3 ((x + 2) (x - 2))}{(x + 2) (x - 2)} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{2 (x - 1) - 3 ((x + 2) (x - 2))}{(x + 2) (x - 2)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Apply the distributive property.

$\frac{2 x + 2 \cdot - 1 - 3 (x + 2) (x - 2)}{(x + 2) (x - 2)} = 0$

Step 2.10.2

Multiply $2$ by $- 1$ .

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

Step 2.10.3

Apply the distributive property.

$\frac{2 x - 2 + (- 3 x - 3 \cdot 2) (x - 2)}{(x + 2) (x - 2)} = 0$

Step 2.10.4

Multiply $- 3$ by $2$ .

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

Step 2.10.5

Expand $(- 3 x - 6) (x - 2)$ using the FOIL Method.

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Step 2.10.5.1

Apply the distributive property.

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

Step 2.10.5.2

Apply the distributive property.

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

Step 2.10.5.3

Apply the distributive property.

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

Step 2.10.6

Simplify and combine like terms.

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Step 2.10.6.1

Simplify each term.

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Step 2.10.6.1.1

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

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Step 2.10.6.1.1.1

Move $x$ .

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

Step 2.10.6.1.1.2

Multiply $x$ by $x$ .

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

Step 2.10.6.1.2

Multiply $- 2$ by $- 3$ .

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

Step 2.10.6.1.3

Multiply $- 6$ by $- 2$ .

$\frac{2 x - 2 - 3 x^{2} + 6 x - 6 x + 12}{(x + 2) (x - 2)} = 0$

Step 2.10.6.2

Subtract $6 x$ from $6 x$ .

$\frac{2 x - 2 - 3 x^{2} + 0 + 12}{(x + 2) (x - 2)} = 0$

Step 2.10.6.3

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

$\frac{2 x - 2 - 3 x^{2} + 12}{(x + 2) (x - 2)} = 0$

Step 2.10.7

Add $- 2$ and $12$ .

$\frac{2 x - 3 x^{2} + 10}{(x + 2) (x - 2)} = 0$

Step 2.10.8

Reorder terms.

$\frac{- 3 x^{2} + 2 x + 10}{(x + 2) (x - 2)} = 0$

Step 2.11

Factor $- 1$ out of $- 3 x^{2}$ .

$\frac{- (3 x^{2}) + 2 x + 10}{(x + 2) (x - 2)} = 0$

Step 2.12

Factor $- 1$ out of $2 x$ .

$\frac{- (3 x^{2}) - (- 2 x) + 10}{(x + 2) (x - 2)} = 0$

Step 2.13

Factor $- 1$ out of $- (3 x^{2}) - (- 2 x)$ .

$\frac{- (3 x^{2} - 2 x) + 10}{(x + 2) (x - 2)} = 0$

Step 2.14

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{- (3 x^{2} - 2 x) - 1 (- 10)}{(x + 2) (x - 2)} = 0$

Step 2.15

Factor $- 1$ out of $- (3 x^{2} - 2 x) - 1 (- 10)$ .

$\frac{- (3 x^{2} - 2 x - 10)}{(x + 2) (x - 2)} = 0$

Step 2.16

Rewrite $- (3 x^{2} - 2 x - 10)$ as $- 1 (3 x^{2} - 2 x - 10)$ .

$\frac{- 1 (3 x^{2} - 2 x - 10)}{(x + 2) (x - 2)} = 0$

Step 2.17

Move the negative in front of the fraction.

$- \frac{3 x^{2} - 2 x - 10}{(x + 2) (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$3 x^{2} - 2 x - 10 = 0$

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 $a = 3$ , $b = - 2$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot - 10)}}{2 \cdot 3}$

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 10}}{2 \cdot 3}$

Step 4.3.1.2

Multiply $- 4 \cdot 3 \cdot - 10$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 10}}{2 \cdot 3}$

Step 4.3.1.2.2

Multiply $- 12$ by $- 10$ .

$x = \frac{2 \pm \sqrt{4 + 120}}{2 \cdot 3}$

Step 4.3.1.3

Add $4$ and $120$ .

$x = \frac{2 \pm \sqrt{124}}{2 \cdot 3}$

Step 4.3.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.3.1.4.1

Factor $4$ out of $124$ .

$x = \frac{2 \pm \sqrt{4 (31)}}{2 \cdot 3}$

Step 4.3.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 3}$

Step 4.3.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{31}}{2 \cdot 3}$

Step 4.3.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{31}}{6}$

Step 4.3.3

Simplify $\frac{2 \pm 2 \sqrt{31}}{6}$ .

$x = \frac{1 \pm \sqrt{31}}{3}$

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 10}}{2 \cdot 3}$

Step 4.4.1.2

Multiply $- 4 \cdot 3 \cdot - 10$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 10}}{2 \cdot 3}$

Step 4.4.1.2.2

Multiply $- 12$ by $- 10$ .

$x = \frac{2 \pm \sqrt{4 + 120}}{2 \cdot 3}$

Step 4.4.1.3

Add $4$ and $120$ .

$x = \frac{2 \pm \sqrt{124}}{2 \cdot 3}$

Step 4.4.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.4.1.4.1

Factor $4$ out of $124$ .

$x = \frac{2 \pm \sqrt{4 (31)}}{2 \cdot 3}$

Step 4.4.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 3}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{31}}{2 \cdot 3}$

Step 4.4.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{31}}{6}$

Step 4.4.3

Simplify $\frac{2 \pm 2 \sqrt{31}}{6}$ .

$x = \frac{1 \pm \sqrt{31}}{3}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{31}}{3}$

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot - 10}}{2 \cdot 3}$

Step 4.5.1.2

Multiply $- 4 \cdot 3 \cdot - 10$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 10}}{2 \cdot 3}$

Step 4.5.1.2.2

Multiply $- 12$ by $- 10$ .

$x = \frac{2 \pm \sqrt{4 + 120}}{2 \cdot 3}$

Step 4.5.1.3

Add $4$ and $120$ .

$x = \frac{2 \pm \sqrt{124}}{2 \cdot 3}$

Step 4.5.1.4

Rewrite $124$ as $2^{2} \cdot 31$ .

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Step 4.5.1.4.1

Factor $4$ out of $124$ .

$x = \frac{2 \pm \sqrt{4 (31)}}{2 \cdot 3}$

Step 4.5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 3}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{31}}{2 \cdot 3}$

Step 4.5.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{31}}{6}$

Step 4.5.3

Simplify $\frac{2 \pm 2 \sqrt{31}}{6}$ .

$x = \frac{1 \pm \sqrt{31}}{3}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{31}}{3}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{31}}{3}, \frac{1 - \sqrt{31}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{31}}{3}, \frac{1 - \sqrt{31}}{3}$

Decimal Form:

$x = 2.18925478 \dots, - 1.52258812 \dots$