Algebra Examples

$\frac{3 x^{2} - 2}{3 x - 2} - \frac{x}{2 - 3 x}$

Step 1

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 2

Factor $- 1$ out of $- 3 x$ .

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

Step 3

Factor $- 1$ out of $- 1 (- 2) - (3 x)$ .

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

Step 4

Reorder terms.

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

Step 5

To write $\frac{3 x^{2} - 2}{3 x - 2}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

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

Step 6

Write each expression with a common denominator of $(3 x - 2) \cdot - 1$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{3 x^{2} - 2}{3 x - 2}$ by $\frac{- 1}{- 1}$ .

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

Step 6.2

Reorder the factors of $(3 x - 2) \cdot - 1$ .

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $- 1$ by $3$ .

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

Step 8.3

Multiply $- 2$ by $- 1$ .

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

Step 8.4

Reorder terms.

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

Step 8.5

Factor by grouping.

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Step 8.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 3 \cdot 2 = - 6$ and whose sum is $b = - 1$ .

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Step 8.5.1.1

Factor $- 1$ out of $- x$ .

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

Step 8.5.1.2

Rewrite $- 1$ as $2$ plus $- 3$

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

Step 8.5.1.3

Apply the distributive property.

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

Step 8.5.2

Factor out the greatest common factor from each group.

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Step 8.5.2.1

Group the first two terms and the last two terms.

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

Step 8.5.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 8.5.3

Factor the polynomial by factoring out the greatest common factor, $- 3 x + 2$ .

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

Step 9

Simplify terms.

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Step 9.1

Cancel the common factor of $- 3 x + 2$ and $3 x - 2$ .

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Step 9.1.1

Factor $- 1$ out of $- 3 x$ .

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

Step 9.1.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 9.1.3

Factor $- 1$ out of $- (3 x) - 1 (- 2)$ .

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

Step 9.1.4

Rewrite $- (3 x - 2)$ as $- 1 (3 x - 2)$ .

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

Step 9.1.5

Cancel the common factor.

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

Step 9.1.6

Rewrite the expression.

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

Step 9.1.7

Move the negative one from the denominator of $\frac{- 1 (x + 1)}{- 1}$ .

$- 1 \cdot (- 1 (x + 1))$

Step 9.2

Rewrite $- 1 \cdot (- 1 (x + 1))$ as $- (- 1 (x + 1))$ .

$- (- 1 (x + 1))$

Step 9.3

Apply the distributive property.

$- (- 1 x - 1 \cdot 1)$

Step 9.4

Simplify the expression.

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Step 9.4.1

Rewrite $- 1 x$ as $- x$ .

$- (- x - 1 \cdot 1)$

Step 9.4.2

Multiply $- 1$ by $1$ .

$- (- x - 1)$

Step 9.5

Apply the distributive property.

$- - x - - 1$

Step 10

Multiply $- - x$ .

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Step 10.1

Multiply $- 1$ by $- 1$ .

$1 x - - 1$

Step 10.2

Multiply $x$ by $1$ .

$x - - 1$

Step 11

Multiply $- 1$ by $- 1$ .

$x + 1$