Algebra Examples

$\frac{1}{36} x^{2} - 25 = 0$

Step 1

Rewrite $\frac{1}{36} x^{2}$ as ${(\frac{1}{6} x)}^{2}$ .

${(\frac{1}{6} x)}^{2} - 25 = 0$

Step 2

Rewrite $25$ as $5^{2}$ .

${(\frac{1}{6} x)}^{2} - 5^{2} = 0$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{6} x$ and $b = 5$ .

$(\frac{1}{6} x + 5) (\frac{1}{6} x - 5) = 0$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{6}$ and $x$ .

$(\frac{x}{6} + 5) (\frac{1}{6} x - 5) = 0$

Step 4.2

Combine $\frac{1}{6}$ and $x$ .

$(\frac{x}{6} + 5) (\frac{x}{6} - 5) = 0$

Step 5

To write $5$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$(\frac{x}{6} + 5 \cdot \frac{6}{6}) (\frac{x}{6} - 5) = 0$

Step 6

Combine $5$ and $\frac{6}{6}$ .

$(\frac{x}{6} + \frac{5 \cdot 6}{6}) (\frac{x}{6} - 5) = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{x + 5 \cdot 6}{6} \cdot (\frac{x}{6} - 5) = 0$

Step 8

Multiply $5$ by $6$ .

$\frac{x + 30}{6} \cdot (\frac{x}{6} - 5) = 0$

Step 9

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{x + 30}{6} \cdot (\frac{x}{6} - 5 \cdot \frac{6}{6}) = 0$

Step 10

Combine $- 5$ and $\frac{6}{6}$ .

$\frac{x + 30}{6} \cdot (\frac{x}{6} + \frac{- 5 \cdot 6}{6}) = 0$

Step 11

Combine the numerators over the common denominator.

$\frac{x + 30}{6} \cdot \frac{x - 5 \cdot 6}{6} = 0$

Step 12

Multiply $- 5$ by $6$ .

$\frac{x + 30}{6} \cdot \frac{x - 30}{6} = 0$

Step 13

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

$\frac{x + 30}{6} = 0$

$\frac{x - 30}{6} = 0$

Step 14

Set $\frac{x + 30}{6}$ equal to $0$ and solve for $x$ .

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Step 14.1

Set $\frac{x + 30}{6}$ equal to $0$ .

$\frac{x + 30}{6} = 0$

Step 14.2

Solve $\frac{x + 30}{6} = 0$ for $x$ .

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Step 14.2.1

Set the numerator equal to zero.

$x + 30 = 0$

Step 14.2.2

Subtract $30$ from both sides of the equation.

$x = - 30$

Step 15

Set $\frac{x - 30}{6}$ equal to $0$ and solve for $x$ .

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Step 15.1

Set $\frac{x - 30}{6}$ equal to $0$ .

$\frac{x - 30}{6} = 0$

Step 15.2

Solve $\frac{x - 30}{6} = 0$ for $x$ .

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Step 15.2.1

Set the numerator equal to zero.

$x - 30 = 0$

Step 15.2.2

Add $30$ to both sides of the equation.

$x = 30$

Step 16

The final solution is all the values that make $\frac{x + 30}{6} \cdot \frac{x - 30}{6} = 0$ true.

$x = - 30, 30$