Pre-Algebra Examples

$15 {(x + 4)}^{2} - 23 {(x + 4)}^{2} = - 4$

Step 1

Subtract $23 {(x + 4)}^{2}$ from $15 {(x + 4)}^{2}$ .

$- 8 {(x + 4)}^{2} = - 4$

Step 2

Divide each term in $- 8 {(x + 4)}^{2} = - 4$ by $- 8$ and simplify.

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Step 2.1

Divide each term in $- 8 {(x + 4)}^{2} = - 4$ by $- 8$ .

$\frac{- 8 {(x + 4)}^{2}}{- 8} = \frac{- 4}{- 8}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 8$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 8 {(x + 4)}^{2}}{- 8} = \frac{- 4}{- 8}$

Step 2.2.1.2

Divide ${(x + 4)}^{2}$ by $1$ .

${(x + 4)}^{2} = \frac{- 4}{- 8}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 4$ and $- 8$ .

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Step 2.3.1.1

Factor $- 4$ out of $- 4$ .

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

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $- 4$ out of $- 8$ .

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

Step 2.3.1.2.2

Cancel the common factor.

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

Step 2.3.1.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

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

Step 4.2

Any root of $1$ is $1$ .

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

Step 4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x + 4 = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.4.4

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

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

Step 4.4.5

Add $1$ and $1$ .

$x + 4 = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.4.6.1

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

$x + 4 = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$x + 4 = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$x + 4 = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$x + 4 = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

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

Step 4.4.6.5

Evaluate the exponent.

$x + 4 = \pm \frac{\sqrt{2}}{2}$

Step 5

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

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Step 5.1

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

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

Step 5.2

Subtract $4$ from both sides of the equation.

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

Step 5.3

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

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

Step 5.4

Subtract $4$ from both sides of the equation.

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

Step 5.5

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

$x = \frac{\sqrt{2}}{2} - 4, - \frac{\sqrt{2}}{2} - 4$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2}}{2} - 4, - \frac{\sqrt{2}}{2} - 4$

Decimal Form:

$x = - 3.29289321 \dots, - 4.70710678 \dots$