Algebra Examples

$2 = \frac{11}{2 x - 1} + \frac{- 5}{{(2 x - 1)}^{2}}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{11}{2 x - 1}$ from both sides of the equation.

$2 - \frac{11}{2 x - 1} = \frac{- 5}{{(2 x - 1)}^{2}}$

Step 1.2

Subtract $\frac{- 5}{{(2 x - 1)}^{2}}$ from both sides of the equation.

$2 - \frac{11}{2 x - 1} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2

Simplify $2 - \frac{11}{2 x - 1} - \frac{- 5}{{(2 x - 1)}^{2}}$ .

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Step 2.1

Find the common denominator.

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Step 2.1.1

Write $2$ as a fraction with denominator $1$ .

$\frac{2}{1} - \frac{11}{2 x - 1} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.2

Multiply $\frac{2}{1}$ by $\frac{{(2 x - 1)}^{2}}{{(2 x - 1)}^{2}}$ .

$\frac{2}{1} \cdot \frac{{(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11}{2 x - 1} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.3

Multiply $\frac{2}{1}$ by $\frac{{(2 x - 1)}^{2}}{{(2 x - 1)}^{2}}$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11}{2 x - 1} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.4

Multiply $\frac{11}{2 x - 1}$ by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - (\frac{11}{2 x - 1} \cdot \frac{2 x - 1}{2 x - 1}) - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.5

Multiply $\frac{11}{2 x - 1}$ by $\frac{2 x - 1}{2 x - 1}$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11 (2 x - 1)}{(2 x - 1) (2 x - 1)} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.6

Raise $2 x - 1$ to the power of $1$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11 (2 x - 1)}{{(2 x - 1)}^{1} (2 x - 1)} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.7

Raise $2 x - 1$ to the power of $1$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11 (2 x - 1)}{{(2 x - 1)}^{1} {(2 x - 1)}^{1}} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11 (2 x - 1)}{{(2 x - 1)}^{1 + 1}} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.1.9

Add $1$ and $1$ .

$\frac{2 {(2 x - 1)}^{2}}{{(2 x - 1)}^{2}} - \frac{11 (2 x - 1)}{{(2 x - 1)}^{2}} - \frac{- 5}{{(2 x - 1)}^{2}} = 0$

Step 2.2

Combine the numerators over the common denominator.

$\frac{2 {(2 x - 1)}^{2} - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3

Simplify each term.

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Step 2.3.1

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

$\frac{2 ((2 x - 1) (2 x - 1)) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.2

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

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Step 2.3.2.1

Apply the distributive property.

$\frac{2 (2 x (2 x - 1) - 1 (2 x - 1)) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.2.2

Apply the distributive property.

$\frac{2 (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.2.3

Apply the distributive property.

$\frac{2 (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 (2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.2

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

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Step 2.3.3.1.2.1

Move $x$ .

$\frac{2 (2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{2 (2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.3

Multiply $2$ by $2$ .

$\frac{2 (4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.4

Multiply $- 1$ by $2$ .

$\frac{2 (4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.5

Multiply $2$ by $- 1$ .

$\frac{2 (4 x^{2} - 2 x - 2 x - 1 \cdot - 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.1.6

Multiply $- 1$ by $- 1$ .

$\frac{2 (4 x^{2} - 2 x - 2 x + 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{2 (4 x^{2} - 4 x + 1) - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.4

Apply the distributive property.

$\frac{2 (4 x^{2}) + 2 (- 4 x) + 2 \cdot 1 - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply $4$ by $2$ .

$\frac{8 x^{2} + 2 (- 4 x) + 2 \cdot 1 - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.5.2

Multiply $- 4$ by $2$ .

$\frac{8 x^{2} - 8 x + 2 \cdot 1 - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.5.3

Multiply $2$ by $1$ .

$\frac{8 x^{2} - 8 x + 2 - 11 (2 x - 1) + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.6

Apply the distributive property.

$\frac{8 x^{2} - 8 x + 2 - 11 (2 x) - 11 \cdot - 1 + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.7

Multiply $2$ by $- 11$ .

$\frac{8 x^{2} - 8 x + 2 - 22 x - 11 \cdot - 1 + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.3.8

Multiply $- 11$ by $- 1$ .

$\frac{8 x^{2} - 8 x + 2 - 22 x + 11 + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.4

Subtract $22 x$ from $- 8 x$ .

$\frac{8 x^{2} - 30 x + 2 + 11 + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.5

Add $2$ and $11$ .

$\frac{8 x^{2} - 30 x + 13 + 5}{{(2 x - 1)}^{2}} = 0$

Step 2.6

Add $13$ and $5$ .

$\frac{8 x^{2} - 30 x + 18}{{(2 x - 1)}^{2}} = 0$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Factor $2$ out of $8 x^{2} - 30 x + 18$ .

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Step 2.7.1.1

Factor $2$ out of $8 x^{2}$ .

$\frac{2 (4 x^{2}) - 30 x + 18}{{(2 x - 1)}^{2}} = 0$

Step 2.7.1.2

Factor $2$ out of $- 30 x$ .

$\frac{2 (4 x^{2}) + 2 (- 15 x) + 18}{{(2 x - 1)}^{2}} = 0$

Step 2.7.1.3

Factor $2$ out of $18$ .

$\frac{2 (4 x^{2}) + 2 (- 15 x) + 2 (9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.1.4

Factor $2$ out of $2 (4 x^{2}) + 2 (- 15 x)$ .

$\frac{2 (4 x^{2} - 15 x) + 2 (9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.1.5

Factor $2$ out of $2 (4 x^{2} - 15 x) + 2 (9)$ .

$\frac{2 (4 x^{2} - 15 x + 9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.2

Factor by grouping.

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Step 2.7.2.1

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 $a \cdot c = 4 \cdot 9 = 36$ and whose sum is $b = - 15$ .

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Step 2.7.2.1.1

Factor $- 15$ out of $- 15 x$ .

$\frac{2 (4 x^{2} - 15 (x) + 9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.2.1.2

Rewrite $- 15$ as $- 3$ plus $- 12$

$\frac{2 (4 x^{2} + (- 3 - 12) x + 9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.2.1.3

Apply the distributive property.

$\frac{2 (4 x^{2} - 3 x - 12 x + 9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.2.2

Factor out the greatest common factor from each group.

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Step 2.7.2.2.1

Group the first two terms and the last two terms.

$\frac{2 ((4 x^{2} - 3 x) - 12 x + 9)}{{(2 x - 1)}^{2}} = 0$

Step 2.7.2.2.2

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

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

Step 2.7.2.3

Factor the polynomial by factoring out the greatest common factor, $4 x - 3$ .

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

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

Step 3

Set the numerator equal to zero.

$2 (4 x - 3) (x - 3) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$4 x - 3 = 0$

$x - 3 = 0$

Step 4.2

Set $4 x - 3$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 4.2.2

Solve $4 x - 3 = 0$ for $x$ .

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Step 4.2.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 4.2.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

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Step 4.2.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2.2

Simplify the left side.

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Step 4.2.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 4.3

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.3.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.4

The final solution is all the values that make $2 (4 x - 3) (x - 3) = 0$ true.

$x = \frac{3}{4}, 3$