Algebra Examples

$\frac{4}{3} {(x + 1)}^{2} + 16 = 0$

Step 1

Subtract $16$ from both sides of the equation.

$\frac{4}{3} \cdot {(x + 1)}^{2} = - 16$

Step 2

Combine $\frac{4}{3}$ and ${(x + 1)}^{2}$ .

$\frac{4 {(x + 1)}^{2}}{3} = - 16$

Step 3

Multiply both sides of the equation by $\frac{3}{4}$ .

$\frac{3}{4} \cdot \frac{4 {(x + 1)}^{2}}{3} = \frac{3}{4} \cdot - 16$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{3}{4} \cdot \frac{4 {(x + 1)}^{2}}{3}$ .

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Step 4.1.1.1

Combine.

$\frac{3 (4 {(x + 1)}^{2})}{4 \cdot 3} = \frac{3}{4} \cdot - 16$

Step 4.1.1.2

Cancel the common factor of $3$ .

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Step 4.1.1.2.1

Cancel the common factor.

$\frac{3 (4 {(x + 1)}^{2})}{4 \cdot 3} = \frac{3}{4} \cdot - 16$

Step 4.1.1.2.2

Rewrite the expression.

$\frac{4 {(x + 1)}^{2}}{4} = \frac{3}{4} \cdot - 16$

Step 4.1.1.3

Cancel the common factor of $4$ .

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Step 4.1.1.3.1

Cancel the common factor.

$\frac{4 {(x + 1)}^{2}}{4} = \frac{3}{4} \cdot - 16$

Step 4.1.1.3.2

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

${(x + 1)}^{2} = \frac{3}{4} \cdot - 16$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{4} \cdot - 16$ .

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Step 4.2.1.1

Cancel the common factor of $4$ .

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Step 4.2.1.1.1

Factor $4$ out of $- 16$ .

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

Step 4.2.1.1.2

Cancel the common factor.

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

Step 4.2.1.1.3

Rewrite the expression.

${(x + 1)}^{2} = 3 \cdot - 4$

Step 4.2.1.2

Multiply $3$ by $- 4$ .

${(x + 1)}^{2} = - 12$

Step 5

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

$x + 1 = \pm \sqrt{- 12}$

Step 6

Simplify $\pm \sqrt{- 12}$ .

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Step 6.1

Rewrite $- 12$ as $- 1 (12)$ .

$x + 1 = \pm \sqrt{- 1 (12)}$

Step 6.2

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x + 1 = \pm \sqrt{- 1} \cdot \sqrt{12}$

Step 6.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x + 1 = \pm i \cdot \sqrt{12}$

Step 6.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 6.4.1

Factor $4$ out of $12$ .

$x + 1 = \pm i \cdot \sqrt{4 (3)}$

Step 6.4.2

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

$x + 1 = \pm i \cdot \sqrt{2^{2} \cdot 3}$

Step 6.5

Pull terms out from under the radical.

$x + 1 = \pm i \cdot (2 \sqrt{3})$

Step 6.6

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

$x + 1 = \pm 2 i \sqrt{3}$

Step 7

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

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Step 7.1

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

$x + 1 = 2 i \sqrt{3}$

Step 7.2

Subtract $1$ from both sides of the equation.

$x = 2 i \sqrt{3} - 1$

Step 7.3

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

$x + 1 = - 2 i \sqrt{3}$

Step 7.4

Subtract $1$ from both sides of the equation.

$x = - 2 i \sqrt{3} - 1$

Step 7.5

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

$x = 2 i \sqrt{3} - 1, - 2 i \sqrt{3} - 1$