Algebra Examples

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

Step 1

Subtract $1$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $x (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{3}{x}$ by $\frac{x - 2}{x - 2}$ .

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

Step 2.3.2

Multiply $\frac{3}{x - 2}$ by $\frac{x}{x}$ .

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

Step 2.3.3

Reorder the factors of $(x - 2) x$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Factor $3$ out of $3 x$ .

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

Step 2.5.1.2

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

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

Step 2.5.2

Add $x$ and $x$ .

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

Step 2.5.3

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

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Step 2.5.3.1

Factor $2$ out of $2 x$ .

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

Step 2.5.3.2

Factor $2$ out of $- 2$ .

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

Step 2.5.3.3

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

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

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

Step 2.5.4

Multiply $3$ by $2$ .

$\frac{6 (x - 1)}{x (x - 2)} - 1 = 0$

Step 2.6

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

$\frac{6 (x - 1)}{x (x - 2)} - 1 \cdot \frac{x (x - 2)}{x (x - 2)} = 0$

Step 2.7

Combine $- 1$ and $\frac{x (x - 2)}{x (x - 2)}$ .

$\frac{6 (x - 1)}{x (x - 2)} + \frac{- (x (x - 2))}{x (x - 2)} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{6 (x - 1) - (x (x - 2))}{x (x - 2)} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

Apply the distributive property.

$\frac{6 x + 6 \cdot - 1 - x (x - 2)}{x (x - 2)} = 0$

Step 2.9.2

Multiply $6$ by $- 1$ .

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

Step 2.9.3

Apply the distributive property.

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

Step 2.9.4

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

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Step 2.9.4.1

Move $x$ .

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

Step 2.9.4.2

Multiply $x$ by $x$ .

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

Step 2.9.5

Multiply $- 2$ by $- 1$ .

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

Step 2.9.6

Add $6 x$ and $2 x$ .

$\frac{8 x - 6 - x^{2}}{x (x - 2)} = 0$

Step 2.9.7

Reorder terms.

$\frac{- x^{2} + 8 x - 6}{x (x - 2)} = 0$

Step 2.10

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

$\frac{- (x^{2}) + 8 x - 6}{x (x - 2)} = 0$

Step 2.11

Factor $- 1$ out of $8 x$ .

$\frac{- (x^{2}) - (- 8 x) - 6}{x (x - 2)} = 0$

Step 2.12

Factor $- 1$ out of $- (x^{2}) - (- 8 x)$ .

$\frac{- (x^{2} - 8 x) - 6}{x (x - 2)} = 0$

Step 2.13

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (x^{2} - 8 x) - 1 (6)}{x (x - 2)} = 0$

Step 2.14

Factor $- 1$ out of $- (x^{2} - 8 x) - 1 (6)$ .

$\frac{- (x^{2} - 8 x + 6)}{x (x - 2)} = 0$

Step 2.15

Rewrite $- (x^{2} - 8 x + 6)$ as $- 1 (x^{2} - 8 x + 6)$ .

$\frac{- 1 (x^{2} - 8 x + 6)}{x (x - 2)} = 0$

Step 2.16

Move the negative in front of the fraction.

$- \frac{x^{2} - 8 x + 6}{x (x - 2)} = 0$

Step 3

Set the numerator equal to zero.

$x^{2} - 8 x + 6 = 0$

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 $a = 1$ , $b = - 8$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 6)}}{2 \cdot 1}$

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 $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 1}$

Step 4.3.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.3.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 4.3.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 4.3.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{10}}{2}$

Step 4.3.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{2}$ .

$x = 4 \pm \sqrt{10}$

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 $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 1}$

Step 4.4.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.4.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 4.4.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{10}}{2}$

Step 4.4.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{2}$ .

$x = 4 \pm \sqrt{10}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = 4 + \sqrt{10}$

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 $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 6$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 6}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $6$ .

$x = \frac{8 \pm \sqrt{64 - 24}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $24$ from $64$ .

$x = \frac{8 \pm \sqrt{40}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 4.5.1.4.1

Factor $4$ out of $40$ .

$x = \frac{8 \pm \sqrt{4 (10)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$x = \frac{8 \pm \sqrt{2^{2} \cdot 10}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{8 \pm 2 \sqrt{10}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 2 \sqrt{10}}{2}$

Step 4.5.3

Simplify $\frac{8 \pm 2 \sqrt{10}}{2}$ .

$x = 4 \pm \sqrt{10}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = 4 - \sqrt{10}$

Step 4.6

The final answer is the combination of both solutions.

$x = 4 + \sqrt{10}, 4 - \sqrt{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 4 + \sqrt{10}, 4 - \sqrt{10}$

Decimal Form:

$x = 7.16227766 \dots, 0.83772233 \dots$