Algebra Examples

$\frac{9 x^{2}}{3 x - 1} + \frac{1}{1 - 3 x}$

Step 1

Simplify terms.

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Step 1.1

Rewrite

$1$ as

$- 1 (- 1)$ .

$\frac{9 x^{2}}{3 x - 1} + \frac{1}{- 1 (- 1) - 3 x}$

Step 1.2

Factor

$- 1$ out of

$- 3 x$ .

$\frac{9 x^{2}}{3 x - 1} + \frac{1}{- 1 (- 1) - (3 x)}$

Step 1.3

Factor

$- 1$ out of

$- 1 (- 1) - (3 x)$ .

$\frac{9 x^{2}}{3 x - 1} + \frac{1}{- 1 (- 1 + 3 x)}$

Step 1.4

Simplify the expression.

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Step 1.4.1

Move a negative from the denominator of

$\frac{1}{- 1 (- 1 + 3 x)}$ to the numerator.

$\frac{9 x^{2}}{3 x - 1} + \frac{- 1 \cdot 1}{- 1 + 3 x}$

Step 1.4.2

Reorder terms.

$\frac{9 x^{2}}{3 x - 1} + \frac{- 1 \cdot 1}{3 x - 1}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{9 x^{2} - 1 \cdot 1}{3 x - 1}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

$9 x^{2}$ as

${(3 x)}^{2}$ .

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

Step 2.2

Rewrite

$1$ as

$1^{2}$ .

$\frac{{(3 x)}^{2} - 1^{2}}{3 x - 1}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 3 x$ and

$b = 1$ .

$\frac{(3 x + 1) (3 x - 1)}{3 x - 1}$

Step 3

Cancel the common factor of

$3 x - 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(3 x + 1) (3 x - 1)}{3 x - 1}$

Step 3.2

Divide

$3 x + 1$ by

$1$ .

$3 x + 1$