Algebra Examples

$\frac{3 x^{2}}{4} = 27$

Step 1

Subtract $27$ from both sides of the equation.

$\frac{3 x^{2}}{4} - 27 = 0$

Step 2

Factor $3$ out of $\frac{3 x^{2}}{4} - 27$ .

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Step 2.1

Factor $3$ out of $\frac{3 x^{2}}{4}$ .

$3 (\frac{x^{2}}{4}) - 27 = 0$

Step 2.2

Factor $3$ out of $- 27$ .

$3 (\frac{x^{2}}{4}) + 3 \cdot - 9 = 0$

Step 2.3

Factor $3$ out of $3 \frac{x^{2}}{4} + 3 \cdot - 9$ .

$3 (\frac{x^{2}}{4} - 9) = 0$

Step 3

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

$3 (\frac{x^{2}}{2^{2}} - 9) = 0$

Step 4

Rewrite $\frac{x^{2}}{2^{2}}$ as ${(\frac{x}{2})}^{2}$ .

$3 ({(\frac{x}{2})}^{2} - 9) = 0$

Step 5

Rewrite $9$ as $3^{2}$ .

$3 ({(\frac{x}{2})}^{2} - 3^{2}) = 0$

Step 6

Factor.

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Step 6.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{2}$ and $b = 3$ .

$3 ((\frac{x}{2} + 3) (\frac{x}{2} - 3)) = 0$

Step 6.2

Remove unnecessary parentheses.

$3 (\frac{x}{2} + 3) (\frac{x}{2} - 3) = 0$

Step 7

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 (\frac{x}{2} + 3 \cdot \frac{2}{2}) (\frac{x}{2} - 3) = 0$

Step 8

Combine $3$ and $\frac{2}{2}$ .

$3 (\frac{x}{2} + \frac{3 \cdot 2}{2}) (\frac{x}{2} - 3) = 0$

Step 9

Combine the numerators over the common denominator.

$3 (\frac{x + 3 \cdot 2}{2}) \cdot (\frac{x}{2} - 3) = 0$

Step 10

Multiply $3$ by $2$ .

$3 (\frac{x + 6}{2}) \cdot (\frac{x}{2} - 3) = 0$

Step 11

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 (\frac{x + 6}{2}) \cdot (\frac{x}{2} - 3 \cdot \frac{2}{2}) = 0$

Step 12

Combine $- 3$ and $\frac{2}{2}$ .

$3 (\frac{x + 6}{2}) \cdot (\frac{x}{2} + \frac{- 3 \cdot 2}{2}) = 0$

Step 13

Combine the numerators over the common denominator.

$3 (\frac{x + 6}{2}) \cdot \frac{x - 3 \cdot 2}{2} = 0$

Step 14

Multiply $- 3$ by $2$ .

$3 (\frac{x + 6}{2}) \cdot \frac{x - 6}{2} = 0$

Step 15

Combine exponents.

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Step 15.1

Combine $3$ and $\frac{x + 6}{2}$ .

$\frac{3 (x + 6)}{2} \cdot \frac{x - 6}{2} = 0$

Step 15.2

Multiply $\frac{3 (x + 6)}{2}$ by $\frac{x - 6}{2}$ .

$\frac{3 (x + 6) (x - 6)}{2 \cdot 2} = 0$

Step 15.3

Multiply $2$ by $2$ .

$\frac{3 (x + 6) (x - 6)}{4} = 0$

Step 16

Set the numerator equal to zero.

$3 (x + 6) (x - 6) = 0$

Step 17

Solve the equation for $x$ .

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Step 17.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 6 = 0$

$x - 6 = 0$

Step 17.2

Set $x + 6$ equal to $0$ and solve for $x$ .

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Step 17.2.1

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Step 17.2.2

Subtract $6$ from both sides of the equation.

$x = - 6$

Step 17.3

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 17.3.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 17.3.2

Add $6$ to both sides of the equation.

$x = 6$

Step 17.4

The final solution is all the values that make $3 (x + 6) (x - 6) = 0$ true.

$x = - 6, 6$