Algebra Examples

$\frac{2 x}{5} + \frac{2 x^{2} - 1}{10} - \frac{4 x + 1}{6}$

Step 1

Find the common denominator.

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Step 1.1

Multiply $\frac{2 x}{5}$ by $\frac{6}{6}$ .

$\frac{2 x}{5} \cdot \frac{6}{6} + \frac{2 x^{2} - 1}{10} - \frac{4 x + 1}{6}$

Step 1.2

Multiply $\frac{2 x}{5}$ by $\frac{6}{6}$ .

$\frac{2 x \cdot 6}{5 \cdot 6} + \frac{2 x^{2} - 1}{10} - \frac{4 x + 1}{6}$

Step 1.3

Multiply $\frac{2 x^{2} - 1}{10}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 6}{5 \cdot 6} + \frac{2 x^{2} - 1}{10} \cdot \frac{3}{3} - \frac{4 x + 1}{6}$

Step 1.4

Multiply $\frac{2 x^{2} - 1}{10}$ by $\frac{3}{3}$ .

$\frac{2 x \cdot 6}{5 \cdot 6} + \frac{(2 x^{2} - 1) \cdot 3}{10 \cdot 3} - \frac{4 x + 1}{6}$

Step 1.5

Multiply $\frac{4 x + 1}{6}$ by $\frac{5}{5}$ .

$\frac{2 x \cdot 6}{5 \cdot 6} + \frac{(2 x^{2} - 1) \cdot 3}{10 \cdot 3} - (\frac{4 x + 1}{6} \cdot \frac{5}{5})$

Step 1.6

Multiply $\frac{4 x + 1}{6}$ by $\frac{5}{5}$ .

$\frac{2 x \cdot 6}{5 \cdot 6} + \frac{(2 x^{2} - 1) \cdot 3}{10 \cdot 3} - \frac{(4 x + 1) \cdot 5}{6 \cdot 5}$

Step 1.7

Multiply $5$ by $6$ .

$\frac{2 x \cdot 6}{30} + \frac{(2 x^{2} - 1) \cdot 3}{10 \cdot 3} - \frac{(4 x + 1) \cdot 5}{6 \cdot 5}$

Step 1.8

Reorder the factors of $10 \cdot 3$ .

$\frac{2 x \cdot 6}{30} + \frac{(2 x^{2} - 1) \cdot 3}{3 \cdot 10} - \frac{(4 x + 1) \cdot 5}{6 \cdot 5}$

Step 1.9

Multiply $3$ by $10$ .

$\frac{2 x \cdot 6}{30} + \frac{(2 x^{2} - 1) \cdot 3}{30} - \frac{(4 x + 1) \cdot 5}{6 \cdot 5}$

Step 1.10

Reorder the factors of $6 \cdot 5$ .

$\frac{2 x \cdot 6}{30} + \frac{(2 x^{2} - 1) \cdot 3}{30} - \frac{(4 x + 1) \cdot 5}{5 \cdot 6}$

Step 1.11

Multiply $5$ by $6$ .

$\frac{2 x \cdot 6}{30} + \frac{(2 x^{2} - 1) \cdot 3}{30} - \frac{(4 x + 1) \cdot 5}{30}$

Step 2

Combine the numerators over the common denominator.

$\frac{2 x \cdot 6 + (2 x^{2} - 1) \cdot 3 - (4 x + 1) \cdot 5}{30}$

Step 3

Simplify each term.

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Step 3.1

Multiply $6$ by $2$ .

$\frac{12 x + (2 x^{2} - 1) \cdot 3 - (4 x + 1) \cdot 5}{30}$

Step 3.2

Apply the distributive property.

$\frac{12 x + 2 x^{2} \cdot 3 - 1 \cdot 3 - (4 x + 1) \cdot 5}{30}$

Step 3.3

Multiply $3$ by $2$ .

$\frac{12 x + 6 x^{2} - 1 \cdot 3 - (4 x + 1) \cdot 5}{30}$

Step 3.4

Multiply $- 1$ by $3$ .

$\frac{12 x + 6 x^{2} - 3 - (4 x + 1) \cdot 5}{30}$

Step 3.5

Apply the distributive property.

$\frac{12 x + 6 x^{2} - 3 + (- (4 x) - 1 \cdot 1) \cdot 5}{30}$

Step 3.6

Multiply $4$ by $- 1$ .

$\frac{12 x + 6 x^{2} - 3 + (- 4 x - 1 \cdot 1) \cdot 5}{30}$

Step 3.7

Multiply $- 1$ by $1$ .

$\frac{12 x + 6 x^{2} - 3 + (- 4 x - 1) \cdot 5}{30}$

Step 3.8

Apply the distributive property.

$\frac{12 x + 6 x^{2} - 3 - 4 x \cdot 5 - 1 \cdot 5}{30}$

Step 3.9

Multiply $5$ by $- 4$ .

$\frac{12 x + 6 x^{2} - 3 - 20 x - 1 \cdot 5}{30}$

Step 3.10

Multiply $- 1$ by $5$ .

$\frac{12 x + 6 x^{2} - 3 - 20 x - 5}{30}$

Step 4

Simplify terms.

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Step 4.1

Subtract $20 x$ from $12 x$ .

$\frac{- 8 x + 6 x^{2} - 3 - 5}{30}$

Step 4.2

Simplify the expression.

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Step 4.2.1

Subtract $5$ from $- 3$ .

$\frac{- 8 x + 6 x^{2} - 8}{30}$

Step 4.2.2

Reorder $- 8 x$ and $6 x^{2}$ .

$\frac{6 x^{2} - 8 x - 8}{30}$

Step 4.3

Cancel the common factor of $6 x^{2} - 8 x - 8$ and $30$ .

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Step 4.3.1

Factor $2$ out of $6 x^{2}$ .

$\frac{2 (3 x^{2}) - 8 x - 8}{30}$

Step 4.3.2

Factor $2$ out of $- 8 x$ .

$\frac{2 (3 x^{2}) + 2 (- 4 x) - 8}{30}$

Step 4.3.3

Factor $2$ out of $2 (3 x^{2}) + 2 (- 4 x)$ .

$\frac{2 (3 x^{2} - 4 x) - 8}{30}$

Step 4.3.4

Factor $2$ out of $- 8$ .

$\frac{2 (3 x^{2} - 4 x) + 2 (- 4)}{30}$

Step 4.3.5

Factor $2$ out of $2 (3 x^{2} - 4 x) + 2 (- 4)$ .

$\frac{2 (3 x^{2} - 4 x - 4)}{30}$

Step 4.3.6

Cancel the common factors.

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Step 4.3.6.1

Factor $2$ out of $30$ .

$\frac{2 (3 x^{2} - 4 x - 4)}{2 (15)}$

Step 4.3.6.2

Cancel the common factor.

$\frac{2 (3 x^{2} - 4 x - 4)}{2 \cdot 15}$

Step 4.3.6.3

Rewrite the expression.

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

Step 5

Factor by grouping.

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Step 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 $a \cdot c = 3 \cdot - 4 = - 12$ and whose sum is $b = - 4$ .

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Step 5.1.1

Factor $- 4$ out of $- 4 x$ .

$\frac{3 x^{2} - 4 (x) - 4}{15}$

Step 5.1.2

Rewrite $- 4$ as $2$ plus $- 6$

$\frac{3 x^{2} + (2 - 6) x - 4}{15}$

Step 5.1.3

Apply the distributive property.

$\frac{3 x^{2} + 2 x - 6 x - 4}{15}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 2 x) - 6 x - 4}{15}$

Step 5.2.2

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

$\frac{x (3 x + 2) - 2 (3 x + 2)}{15}$

Step 5.3

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

$\frac{(3 x + 2) (x - 2)}{15}$