Algebra Examples

$x - 8 = - \sqrt{x^{2} - 8}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \sqrt{x^{2} - 8} = x - 8$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${(- \sqrt{x^{2} - 8})}^{2} = {(x - 8)}^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 8}$ as ${(x^{2} - 8)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${(- {(x^{2} - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Apply the product rule to $- {(x^{2} - 8)}^{\frac{1}{2}}$ .

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

Step 3.2.1.2

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

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

Step 3.2.1.3

Multiply ${({(x^{2} - 8)}^{\frac{1}{2}})}^{2}$ by $1$ .

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

Step 3.2.1.4

Multiply the exponents in ${({(x^{2} - 8)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2.1.4.2

Cancel the common factor of $2$ .

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Step 3.2.1.4.2.1

Cancel the common factor.

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

Step 3.2.1.4.2.2

Rewrite the expression.

${(x^{2} - 8)}^{1} = {(x - 8)}^{2}$

Step 3.2.1.5

Simplify.

$x^{2} - 8 = {(x - 8)}^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${(x - 8)}^{2}$ .

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Step 3.3.1.1

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

$x^{2} - 8 = (x - 8) (x - 8)$

Step 3.3.1.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 3.3.1.2.1

Apply the distributive property.

$x^{2} - 8 = x (x - 8) - 8 (x - 8)$

Step 3.3.1.2.2

Apply the distributive property.

$x^{2} - 8 = x \cdot x + x \cdot - 8 - 8 (x - 8)$

Step 3.3.1.2.3

Apply the distributive property.

$x^{2} - 8 = x \cdot x + x \cdot - 8 - 8 x - 8 \cdot - 8$

Step 3.3.1.3

Simplify and combine like terms.

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Step 3.3.1.3.1

Simplify each term.

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Step 3.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} - 8 = x^{2} + x \cdot - 8 - 8 x - 8 \cdot - 8$

Step 3.3.1.3.1.2

Move $- 8$ to the left of $x$ .

$x^{2} - 8 = x^{2} - 8 \cdot x - 8 x - 8 \cdot - 8$

Step 3.3.1.3.1.3

Multiply $- 8$ by $- 8$ .

$x^{2} - 8 = x^{2} - 8 x - 8 x + 64$

Step 3.3.1.3.2

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

$x^{2} - 8 = x^{2} - 16 x + 64$

Step 4

Solve for $x$ .

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Step 4.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 16 x + 64 = x^{2} - 8$

Step 4.2

Move all terms containing $x$ to the left side of the equation.

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Step 4.2.1

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 16 x + 64 - x^{2} = - 8$

Step 4.2.2

Combine the opposite terms in $x^{2} - 16 x + 64 - x^{2}$ .

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Step 4.2.2.1

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

$- 16 x + 64 + 0 = - 8$

Step 4.2.2.2

Add $- 16 x + 64$ and $0$ .

$- 16 x + 64 = - 8$

Step 4.3

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

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Step 4.3.1

Subtract $64$ from both sides of the equation.

$- 16 x = - 8 - 64$

Step 4.3.2

Subtract $64$ from $- 8$ .

$- 16 x = - 72$

Step 4.4

Divide each term in $- 16 x = - 72$ by $- 16$ and simplify.

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Step 4.4.1

Divide each term in $- 16 x = - 72$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{- 72}{- 16}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $- 16$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{- 72}{- 16}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 72}{- 16}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Cancel the common factor of $- 72$ and $- 16$ .

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Step 4.4.3.1.1

Factor $- 8$ out of $- 72$ .

$x = \frac{- 8 (9)}{- 16}$

Step 4.4.3.1.2

Cancel the common factors.

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Step 4.4.3.1.2.1

Factor $- 8$ out of $- 16$ .

$x = \frac{- 8 \cdot 9}{- 8 \cdot 2}$

Step 4.4.3.1.2.2

Cancel the common factor.

$x = \frac{- 8 \cdot 9}{- 8 \cdot 2}$

Step 4.4.3.1.2.3

Rewrite the expression.

$x = \frac{9}{2}$

Step 5