Algebra Examples

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

Step 1

Subtract $3$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 2.3

Write each expression with a common denominator of $(x - 3) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

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

Step 2.3.2

Multiply $\frac{1}{x}$ by $\frac{x - 3}{x - 3}$ .

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

Step 2.3.3

Reorder the factors of $(x - 3) x$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

Simplify the numerator.

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Step 2.5.1

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

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Step 2.5.1.1

Move $x$ .

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

Step 2.5.1.2

Multiply $x$ by $x$ .

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

Step 2.5.2

Factor by grouping.

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Step 2.5.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

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Step 2.5.2.1.1

Multiply by $1$ .

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

Step 2.5.2.1.2

Rewrite $1$ as $- 2$ plus $3$

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

Step 2.5.2.1.3

Apply the distributive property.

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

Step 2.5.2.2

Factor out the greatest common factor from each group.

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Step 2.5.2.2.1

Group the first two terms and the last two terms.

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

Step 2.5.2.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 2.5.2.3

Factor the polynomial by factoring out the greatest common factor, $x - 1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

Simplify the numerator.

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Step 2.9.1

Expand $(x - 1) (2 x + 3)$ using the FOIL Method.

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Step 2.9.1.1

Apply the distributive property.

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

Step 2.9.1.2

Apply the distributive property.

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

Step 2.9.1.3

Apply the distributive property.

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

Step 2.9.2

Simplify and combine like terms.

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Step 2.9.2.1

Simplify each term.

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Step 2.9.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.9.2.1.2

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

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Step 2.9.2.1.2.1

Move $x$ .

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

Step 2.9.2.1.2.2

Multiply $x$ by $x$ .

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

Step 2.9.2.1.3

Move $3$ to the left of $x$ .

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

Step 2.9.2.1.4

Multiply $2$ by $- 1$ .

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

Step 2.9.2.1.5

Multiply $- 1$ by $3$ .

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

Step 2.9.2.2

Subtract $2 x$ from $3 x$ .

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

Step 2.9.3

Apply the distributive property.

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

Step 2.9.4

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

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Step 2.9.4.1

Move $x$ .

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

Step 2.9.4.2

Multiply $x$ by $x$ .

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

Step 2.9.5

Multiply $- 3$ by $- 3$ .

$\frac{2 x^{2} + x - 3 - 3 x^{2} + 9 x}{x (x - 3)} = 0$

Step 2.9.6

Subtract $3 x^{2}$ from $2 x^{2}$ .

$\frac{- x^{2} + x - 3 + 9 x}{x (x - 3)} = 0$

Step 2.9.7

Add $x$ and $9 x$ .

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

Step 2.10

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

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

Step 2.11

Factor $- 1$ out of $10 x$ .

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

Step 2.12

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

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

Step 2.13

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 2.14

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

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

Step 2.15

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

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

Step 2.16

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

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

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 $a = 1$ , $b = - 10$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (1 \cdot 3)}}{2 \cdot 1}$

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 $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.3.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.3.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12}}{2 \cdot 1}$

Step 4.3.1.3

Subtract $12$ from $100$ .

$x = \frac{10 \pm \sqrt{88}}{2 \cdot 1}$

Step 4.3.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 4.3.1.4.1

Factor $4$ out of $88$ .

$x = \frac{10 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 4.3.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 4.3.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 4.3.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{22}}{2}$

Step 4.3.3

Simplify $\frac{10 \pm 2 \sqrt{22}}{2}$ .

$x = 5 \pm \sqrt{22}$

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 $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.4.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12}}{2 \cdot 1}$

Step 4.4.1.3

Subtract $12$ from $100$ .

$x = \frac{10 \pm \sqrt{88}}{2 \cdot 1}$

Step 4.4.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 4.4.1.4.1

Factor $4$ out of $88$ .

$x = \frac{10 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 4.4.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 4.4.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{22}}{2}$

Step 4.4.3

Simplify $\frac{10 \pm 2 \sqrt{22}}{2}$ .

$x = 5 \pm \sqrt{22}$

Step 4.4.4

Change the $\pm$ to $+$ .

$x = 5 + \sqrt{22}$

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 $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 1 \cdot 3}}{2 \cdot 1}$

Step 4.5.1.2

Multiply $- 4 \cdot 1 \cdot 3$ .

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Step 4.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 3}}{2 \cdot 1}$

Step 4.5.1.2.2

Multiply $- 4$ by $3$ .

$x = \frac{10 \pm \sqrt{100 - 12}}{2 \cdot 1}$

Step 4.5.1.3

Subtract $12$ from $100$ .

$x = \frac{10 \pm \sqrt{88}}{2 \cdot 1}$

Step 4.5.1.4

Rewrite $88$ as $2^{2} \cdot 22$ .

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Step 4.5.1.4.1

Factor $4$ out of $88$ .

$x = \frac{10 \pm \sqrt{4 (22)}}{2 \cdot 1}$

Step 4.5.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 22}}{2 \cdot 1}$

Step 4.5.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{22}}{2 \cdot 1}$

Step 4.5.2

Multiply $2$ by $1$ .

$x = \frac{10 \pm 2 \sqrt{22}}{2}$

Step 4.5.3

Simplify $\frac{10 \pm 2 \sqrt{22}}{2}$ .

$x = 5 \pm \sqrt{22}$

Step 4.5.4

Change the $\pm$ to $-$ .

$x = 5 - \sqrt{22}$

Step 4.6

The final answer is the combination of both solutions.

$x = 5 + \sqrt{22}, 5 - \sqrt{22}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 5 + \sqrt{22}, 5 - \sqrt{22}$

Decimal Form:

$x = 9.69041575 \dots, 0.30958424 \dots$