Pre-Algebra Examples

${(4 x + 9)}^{2} + 49 = 0$

Step 1

Subtract $49$ from both sides of the equation.

${(4 x + 9)}^{2} = - 49$

Step 2

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

$4 x + 9 = \pm \sqrt{- 49}$

Step 3

Simplify $\pm \sqrt{- 49}$ .

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Step 3.1

Rewrite $- 49$ as $- 1 (49)$ .

$4 x + 9 = \pm \sqrt{- 1 (49)}$

Step 3.2

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$4 x + 9 = \pm \sqrt{- 1} \cdot \sqrt{49}$

Step 3.3

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

$4 x + 9 = \pm i \cdot \sqrt{49}$

Step 3.4

Rewrite $49$ as $7^{2}$ .

$4 x + 9 = \pm i \cdot \sqrt{7^{2}}$

Step 3.5

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

$4 x + 9 = \pm i \cdot 7$

Step 3.6

Move $7$ to the left of $i$ .

$4 x + 9 = \pm 7 i$

Step 4

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

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Step 4.1

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

$4 x + 9 = 7 i$

Step 4.2

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

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Step 4.2.1

Subtract $9$ from both sides of the equation.

$4 x = 7 i - 9$

Step 4.2.2

Reorder $7 i$ and $- 9$ .

$4 x = - 9 + 7 i$

Step 4.3

Divide each term in $4 x = - 9 + 7 i$ by $4$ and simplify.

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Step 4.3.1

Divide each term in $4 x = - 9 + 7 i$ by $4$ .

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{7 i}{4}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $4$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{7 i}{4}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{4} + \frac{7 i}{4}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Move the negative in front of the fraction.

$x = - \frac{9}{4} + \frac{7 i}{4}$

Step 4.4

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

$4 x + 9 = - 7 i$

Step 4.5

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

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Step 4.5.1

Subtract $9$ from both sides of the equation.

$4 x = - 7 i - 9$

Step 4.5.2

Reorder $- 7 i$ and $- 9$ .

$4 x = - 9 - 7 i$

Step 4.6

Divide each term in $4 x = - 9 - 7 i$ by $4$ and simplify.

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Step 4.6.1

Divide each term in $4 x = - 9 - 7 i$ by $4$ .

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{- 7 i}{4}$

Step 4.6.2

Simplify the left side.

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Step 4.6.2.1

Cancel the common factor of $4$ .

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Step 4.6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 9}{4} + \frac{- 7 i}{4}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{4} + \frac{- 7 i}{4}$

Step 4.6.3

Simplify the right side.

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Step 4.6.3.1

Simplify each term.

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Step 4.6.3.1.1

Move the negative in front of the fraction.

$x = - \frac{9}{4} + \frac{- 7 i}{4}$

Step 4.6.3.1.2

Move the negative in front of the fraction.

$x = - \frac{9}{4} - \frac{7 i}{4}$

Step 4.7

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

$x = - \frac{9}{4} + \frac{7 i}{4}, - \frac{9}{4} - \frac{7 i}{4}$