Algebra Examples

$4 {(x - 2)}^{- 2} = 16$

Step 1

Divide each term in $4 {(x - 2)}^{- 2} = 16$ by $4$ and simplify.

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Step 1.1

Divide each term in $4 {(x - 2)}^{- 2} = 16$ by $4$ .

$\frac{4 {(x - 2)}^{- 2}}{4} = \frac{16}{4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Move ${(x - 2)}^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{4 {(x - 2)}^{2}} = \frac{16}{4}$

Step 1.2.2

Cancel the common factor of $4$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{4}{4 {(x - 2)}^{2}} = \frac{16}{4}$

Step 1.2.2.2

Rewrite the expression.

$\frac{1}{{(x - 2)}^{2}} = \frac{16}{4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $16$ by $4$ .

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

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

${(x - 2)}^{2}, 1$

Step 2.2

The LCM of one and any expression is the expression.

${(x - 2)}^{2}$

Step 3

Multiply each term in $\frac{1}{{(x - 2)}^{2}} = 4$ by ${(x - 2)}^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{1}{{(x - 2)}^{2}} = 4$ by ${(x - 2)}^{2}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of ${(x - 2)}^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

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

Step 3.2.1.2

Rewrite the expression.

$1 = 4 {(x - 2)}^{2}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $4 {(x - 2)}^{2} = 1$ .

$4 {(x - 2)}^{2} = 1$

Step 4.2

Divide each term in $4 {(x - 2)}^{2} = 1$ by $4$ and simplify.

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Step 4.2.1

Divide each term in $4 {(x - 2)}^{2} = 1$ by $4$ .

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $4$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Divide ${(x - 2)}^{2}$ by $1$ .

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

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 2 = \pm \sqrt{\frac{1}{4}}$

Step 4.4

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x - 2 = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 4.4.2

Any root of $1$ is $1$ .

$x - 2 = \pm \frac{1}{\sqrt{4}}$

Step 4.4.3

Simplify the denominator.

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Step 4.4.3.1

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

$x - 2 = \pm \frac{1}{\sqrt{2^{2}}}$

Step 4.4.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x - 2 = \pm \frac{1}{2}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

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

Step 4.5.2

Move all terms not containing $x$ to the right side of the equation.

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Step 4.5.2.1

Add $2$ to both sides of the equation.

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

Step 4.5.2.2

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

$x = \frac{1}{2} + 2 \cdot \frac{2}{2}$

Step 4.5.2.3

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

$x = \frac{1}{2} + \frac{2 \cdot 2}{2}$

Step 4.5.2.4

Combine the numerators over the common denominator.

$x = \frac{1 + 2 \cdot 2}{2}$

Step 4.5.2.5

Simplify the numerator.

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Step 4.5.2.5.1

Multiply $2$ by $2$ .

$x = \frac{1 + 4}{2}$

Step 4.5.2.5.2

Add $1$ and $4$ .

$x = \frac{5}{2}$

Step 4.5.3

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 4.5.4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.5.4.1

Add $2$ to both sides of the equation.

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

Step 4.5.4.2

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

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

Step 4.5.4.3

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

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

Step 4.5.4.4

Combine the numerators over the common denominator.

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

Step 4.5.4.5

Simplify the numerator.

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Step 4.5.4.5.1

Multiply $2$ by $2$ .

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

Step 4.5.4.5.2

Add $- 1$ and $4$ .

$x = \frac{3}{2}$

Step 4.5.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{5}{2}, \frac{3}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}, \frac{3}{2}$

Decimal Form:

$x = 2.5, 1.5$

Mixed Number Form:

$x = 2 \frac{1}{2}, 1 \frac{1}{2}$