Algebra Examples

$x^{2} + \frac{1}{2} x + \frac{1}{16} = \frac{4}{9}$

Step 1

Subtract $\frac{4}{9}$ from both sides of the equation.

$x^{2} + \frac{1}{2} x + \frac{1}{16} - \frac{4}{9} = 0$

Step 2

Simplify $x^{2} + \frac{1}{2} x + \frac{1}{16} - \frac{4}{9}$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} - \frac{4}{9} = 0$

Step 2.2

To write $\frac{1}{16}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} \cdot \frac{9}{9} - \frac{4}{9} = 0$

Step 2.3

To write $- \frac{4}{9}$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} \cdot \frac{9}{9} - \frac{4}{9} \cdot \frac{16}{16} = 0$

Step 2.4

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{16}$ by $\frac{9}{9}$ .

$x^{2} + \frac{x}{2} + \frac{9}{16 \cdot 9} - \frac{4}{9} \cdot \frac{16}{16} = 0$

Step 2.4.2

Multiply $16$ by $9$ .

$x^{2} + \frac{x}{2} + \frac{9}{144} - \frac{4}{9} \cdot \frac{16}{16} = 0$

Step 2.4.3

Multiply $\frac{4}{9}$ by $\frac{16}{16}$ .

$x^{2} + \frac{x}{2} + \frac{9}{144} - \frac{4 \cdot 16}{9 \cdot 16} = 0$

Step 2.4.4

Multiply $9$ by $16$ .

$x^{2} + \frac{x}{2} + \frac{9}{144} - \frac{4 \cdot 16}{144} = 0$

Step 2.5

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{2} + \frac{9 - 4 \cdot 16}{144} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 4$ by $16$ .

$x^{2} + \frac{x}{2} + \frac{9 - 64}{144} = 0$

Step 2.6.2

Subtract $64$ from $9$ .

$x^{2} + \frac{x}{2} + \frac{- 55}{144} = 0$

Step 2.7

Move the negative in front of the fraction.

$x^{2} + \frac{x}{2} - \frac{55}{144} = 0$

Step 3

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

$x^{2} \cdot \frac{2}{2} + \frac{x}{2} - \frac{55}{144} = 0$

Step 4

Combine $x^{2}$ and $\frac{2}{2}$ .

$\frac{x^{2} \cdot 2}{2} + \frac{x}{2} - \frac{55}{144} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 2 + x}{2} - \frac{55}{144} = 0$

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $x^{2} \cdot 2 + x$ .

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Step 6.1.1

Factor $x$ out of $x^{2} \cdot 2$ .

$\frac{x (x \cdot 2) + x}{2} - \frac{55}{144} = 0$

Step 6.1.2

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

$\frac{x (x \cdot 2) + x}{2} - \frac{55}{144} = 0$

Step 6.1.3

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

$\frac{x (x \cdot 2) + x \cdot 1}{2} - \frac{55}{144} = 0$

Step 6.1.4

Factor $x$ out of $x (x \cdot 2) + x \cdot 1$ .

$\frac{x (x \cdot 2 + 1)}{2} - \frac{55}{144} = 0$

Step 6.2

Move $2$ to the left of $x$ .

$\frac{x (2 x + 1)}{2} - \frac{55}{144} = 0$

Step 7

To write $\frac{x (2 x + 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{72}{72}$ .

$\frac{x (2 x + 1)}{2} \cdot \frac{72}{72} - \frac{55}{144} = 0$

Step 8

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.1

Multiply $\frac{x (2 x + 1)}{2}$ by $\frac{72}{72}$ .

$\frac{x (2 x + 1) \cdot 72}{2 \cdot 72} - \frac{55}{144} = 0$

Step 8.2

Multiply $2$ by $72$ .

$\frac{x (2 x + 1) \cdot 72}{144} - \frac{55}{144} = 0$

Step 9

Combine the numerators over the common denominator.

$\frac{x (2 x + 1) \cdot 72 - 55}{144} = 0$

Step 10

Simplify the numerator.

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Step 10.1

Apply the distributive property.

$\frac{(x (2 x) + x \cdot 1) \cdot 72 - 55}{144} = 0$

Step 10.2

Rewrite using the commutative property of multiplication.

$\frac{(2 x \cdot x + x \cdot 1) \cdot 72 - 55}{144} = 0$

Step 10.3

Multiply $x$ by $1$ .

$\frac{(2 x \cdot x + x) \cdot 72 - 55}{144} = 0$

Step 10.4

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

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Step 10.4.1

Move $x$ .

$\frac{(2 (x \cdot x) + x) \cdot 72 - 55}{144} = 0$

Step 10.4.2

Multiply $x$ by $x$ .

$\frac{(2 x^{2} + x) \cdot 72 - 55}{144} = 0$

Step 10.5

Apply the distributive property.

$\frac{2 x^{2} \cdot 72 + x \cdot 72 - 55}{144} = 0$

Step 10.6

Multiply $72$ by $2$ .

$\frac{144 x^{2} + x \cdot 72 - 55}{144} = 0$

Step 10.7

Move $72$ to the left of $x$ .

$\frac{144 x^{2} + 72 x - 55}{144} = 0$

Step 10.8

Factor by grouping.

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Step 10.8.1

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 $a \cdot c = 144 \cdot - 55 = - 7920$ and whose sum is $b = 72$ .

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Step 10.8.1.1

Factor $72$ out of $72 x$ .

$\frac{144 x^{2} + 72 (x) - 55}{144} = 0$

Step 10.8.1.2

Rewrite $72$ as $- 60$ plus $132$

$\frac{144 x^{2} + (- 60 + 132) x - 55}{144} = 0$

Step 10.8.1.3

Apply the distributive property.

$\frac{144 x^{2} - 60 x + 132 x - 55}{144} = 0$

Step 10.8.2

Factor out the greatest common factor from each group.

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Step 10.8.2.1

Group the first two terms and the last two terms.

$\frac{(144 x^{2} - 60 x) + 132 x - 55}{144} = 0$

Step 10.8.2.2

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

$\frac{12 x (12 x - 5) + 11 (12 x - 5)}{144} = 0$

Step 10.8.3

Factor the polynomial by factoring out the greatest common factor, $12 x - 5$ .

$\frac{(12 x - 5) (12 x + 11)}{144} = 0$

Step 11

Set the numerator equal to zero.

$(12 x - 5) (12 x + 11) = 0$

Step 12

Solve the equation for $x$ .

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Step 12.1

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

$12 x - 5 = 0$

$12 x + 11 = 0$

Step 12.2

Set $12 x - 5$ equal to $0$ and solve for $x$ .

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Step 12.2.1

Set $12 x - 5$ equal to $0$ .

$12 x - 5 = 0$

Step 12.2.2

Solve $12 x - 5 = 0$ for $x$ .

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Step 12.2.2.1

Add $5$ to both sides of the equation.

$12 x = 5$

Step 12.2.2.2

Divide each term in $12 x = 5$ by $12$ and simplify.

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Step 12.2.2.2.1

Divide each term in $12 x = 5$ by $12$ .

$\frac{12 x}{12} = \frac{5}{12}$

Step 12.2.2.2.2

Simplify the left side.

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Step 12.2.2.2.2.1

Cancel the common factor of $12$ .

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Step 12.2.2.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{5}{12}$

Step 12.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5}{12}$

Step 12.3

Set $12 x + 11$ equal to $0$ and solve for $x$ .

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Step 12.3.1

Set $12 x + 11$ equal to $0$ .

$12 x + 11 = 0$

Step 12.3.2

Solve $12 x + 11 = 0$ for $x$ .

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Step 12.3.2.1

Subtract $11$ from both sides of the equation.

$12 x = - 11$

Step 12.3.2.2

Divide each term in $12 x = - 11$ by $12$ and simplify.

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Step 12.3.2.2.1

Divide each term in $12 x = - 11$ by $12$ .

$\frac{12 x}{12} = \frac{- 11}{12}$

Step 12.3.2.2.2

Simplify the left side.

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Step 12.3.2.2.2.1

Cancel the common factor of $12$ .

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Step 12.3.2.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{- 11}{12}$

Step 12.3.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 11}{12}$

Step 12.3.2.2.3

Simplify the right side.

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Step 12.3.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{11}{12}$

Step 12.4

The final solution is all the values that make $(12 x - 5) (12 x + 11) = 0$ true.

$x = \frac{5}{12}, - \frac{11}{12}$