Algebra Examples

$- \frac{4}{3} x^{2} - 15 = 201$

Step 1

Simplify each term.

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Step 1.1

Combine $x^{2}$ and $\frac{4}{3}$ .

$- \frac{x^{2} \cdot 4}{3} - 15 = 201$

Step 1.2

Move $4$ to the left of $x^{2}$ .

$- \frac{4 x^{2}}{3} - 15 = 201$

Step 2

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

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Step 2.1

Add $15$ to both sides of the equation.

$- \frac{4 x^{2}}{3} = 201 + 15$

Step 2.2

Add $201$ and $15$ .

$- \frac{4 x^{2}}{3} = 216$

Step 3

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

$- \frac{3}{4} (- \frac{4 x^{2}}{3}) = - \frac{3}{4} \cdot 216$

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 $- \frac{3}{4} (- \frac{4 x^{2}}{3})$ .

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (- \frac{4 x^{2}}{3}) = - \frac{3}{4} \cdot 216$

Step 4.1.1.1.2

Move the leading negative in $- \frac{4 x^{2}}{3}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 4 x^{2}}{3} = - \frac{3}{4} \cdot 216$

Step 4.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 4 x^{2}}{3} = - \frac{3}{4} \cdot 216$

Step 4.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 4 x^{2}}{3} = - \frac{3}{4} \cdot 216$

Step 4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 x^{2}) = - \frac{3}{4} \cdot 216$

Step 4.1.1.2

Cancel the common factor of $4$ .

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Step 4.1.1.2.1

Factor $4$ out of $- 4 x^{2}$ .

$\frac{- 1}{4} (4 (- x^{2})) = - \frac{3}{4} \cdot 216$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- x^{2})) = - \frac{3}{4} \cdot 216$

Step 4.1.1.2.3

Rewrite the expression.

$- - x^{2} = - \frac{3}{4} \cdot 216$

Step 4.1.1.3

Multiply.

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Step 4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - \frac{3}{4} \cdot 216$

Step 4.1.1.3.2

Multiply $x^{2}$ by $1$ .

$x^{2} = - \frac{3}{4} \cdot 216$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{3}{4} \cdot 216$ .

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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

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$x^{2} = \frac{- 3}{4} \cdot 216$

Step 4.2.1.1.2

Factor $4$ out of $216$ .

$x^{2} = \frac{- 3}{4} \cdot (4 (54))$

Step 4.2.1.1.3

Cancel the common factor.

$x^{2} = \frac{- 3}{4} \cdot (4 \cdot 54)$

Step 4.2.1.1.4

Rewrite the expression.

$x^{2} = - 3 \cdot 54$

Step 4.2.1.2

Multiply $- 3$ by $54$ .

$x^{2} = - 162$

Step 5

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

$x = \pm \sqrt{- 162}$

Step 6

Simplify $\pm \sqrt{- 162}$ .

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Step 6.1

Rewrite $- 162$ as $- 1 (162)$ .

$x = \pm \sqrt{- 1 (162)}$

Step 6.2

Rewrite $\sqrt{- 1 (162)}$ as $\sqrt{- 1} \cdot \sqrt{162}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{162}$

Step 6.3

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

$x = \pm i \cdot \sqrt{162}$

Step 6.4

Rewrite $162$ as $9^{2} \cdot 2$ .

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Step 6.4.1

Factor $81$ out of $162$ .

$x = \pm i \cdot \sqrt{81 (2)}$

Step 6.4.2

Rewrite $81$ as $9^{2}$ .

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

Step 6.5

Pull terms out from under the radical.

$x = \pm i \cdot (9 \sqrt{2})$

Step 6.6

Move $9$ to the left of $i$ .

$x = \pm 9 i \sqrt{2}$

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.

$x = 9 i \sqrt{2}$

Step 7.2

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

$x = - 9 i \sqrt{2}$

Step 7.3

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

$x = 9 i \sqrt{2}, - 9 i \sqrt{2}$