Algebra Examples

$\frac{2 a^{2} - 7 a - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \div \frac{a^{2} + 3 a - 10}{2 a^{2} - 9 a + 9}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 a^{2} - 7 a - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = - 7$ .

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Step 2.1.1

Factor $- 7$ out of $- 7 a$ .

$\frac{2 a^{2} - 7 (a) - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2.1.2

Rewrite $- 7$ as $3$ plus $- 10$

$\frac{2 a^{2} + (3 - 10) a - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2.1.3

Apply the distributive property.

$\frac{2 a^{2} + 3 a - 10 a - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(2 a^{2} + 3 a) - 10 a - 15}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2.2.2

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

$\frac{a (2 a + 3) - 5 (2 a + 3)}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 2.3

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

$\frac{(2 a + 3) (a - 5)}{3 a^{2} - 8 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 3 \cdot - 3 = - 9$ and whose sum is $b = - 8$ .

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Step 3.1.1

Factor $- 8$ out of $- 8 a$ .

$\frac{(2 a + 3) (a - 5)}{3 a^{2} - 8 (a) - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.1.2

Rewrite $- 8$ as $1$ plus $- 9$

$\frac{(2 a + 3) (a - 5)}{3 a^{2} + (1 - 9) a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.1.3

Apply the distributive property.

$\frac{(2 a + 3) (a - 5)}{3 a^{2} + 1 a - 9 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.1.4

Multiply $a$ by $1$ .

$\frac{(2 a + 3) (a - 5)}{3 a^{2} + a - 9 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(2 a + 3) (a - 5)}{(3 a^{2} + a) - 9 a - 3} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.2.2

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

$\frac{(2 a + 3) (a - 5)}{a (3 a + 1) - 3 (3 a + 1)} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 a + 1$ .

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{9 a^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $9 a^{2}$ as ${(3 a)}^{2}$ .

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{{(3 a)}^{2} - 1}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 4.2

Rewrite $1$ as $1^{2}$ .

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{{(3 a)}^{2} - 1^{2}}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 a$ and $b = 1$ .

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{4 a^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $4 a^{2}$ as ${(2 a)}^{2}$ .

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{{(2 a)}^{2} - 9} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 5.2

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

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{{(2 a)}^{2} - 3^{2}} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 5.3

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

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{(2 a + 3) (2 a - 3)} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of $2 a + 3$ .

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Step 6.1.1

Cancel the common factor.

$\frac{(2 a + 3) (a - 5)}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{(2 a + 3) (2 a - 3)} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 6.1.2

Rewrite the expression.

$\frac{a - 5}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{2 a - 3} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 6.2

Cancel the common factor of $3 a + 1$ .

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Step 6.2.1

Cancel the common factor.

$\frac{a - 5}{(3 a + 1) (a - 3)} \cdot \frac{(3 a + 1) (3 a - 1)}{2 a - 3} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 6.2.2

Rewrite the expression.

$\frac{a - 5}{a - 3} \cdot \frac{3 a - 1}{2 a - 3} \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 6.3

Multiply $\frac{a - 5}{a - 3}$ by $\frac{3 a - 1}{2 a - 3}$ .

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{2 a^{2} - 9 a + 9}{a^{2} + 3 a - 10}$

Step 7

Factor by grouping.

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Step 7.1

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 $a \cdot c = 2 \cdot 9 = 18$ and whose sum is $b = - 9$ .

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Step 7.1.1

Factor $- 9$ out of $- 9 a$ .

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{2 a^{2} - 9 (a) + 9}{a^{2} + 3 a - 10}$

Step 7.1.2

Rewrite $- 9$ as $- 3$ plus $- 6$

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{2 a^{2} + (- 3 - 6) a + 9}{a^{2} + 3 a - 10}$

Step 7.1.3

Apply the distributive property.

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{2 a^{2} - 3 a - 6 a + 9}{a^{2} + 3 a - 10}$

Step 7.2

Factor out the greatest common factor from each group.

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Step 7.2.1

Group the first two terms and the last two terms.

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{(2 a^{2} - 3 a) - 6 a + 9}{a^{2} + 3 a - 10}$

Step 7.2.2

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

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

Step 7.3

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

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

Step 8

Factor $a^{2} + 3 a - 10$ using the AC method.

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Step 8.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 8.2

Write the factored form using these integers.

$\frac{(a - 5) (3 a - 1)}{(a - 3) (2 a - 3)} \cdot \frac{(2 a - 3) (a - 3)}{(a - 2) (a + 5)}$

Step 9

Cancel the common factor of $(2 a - 3) (a - 3)$ .

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Step 9.1

Factor $(2 a - 3) (a - 3)$ out of $(a - 3) (2 a - 3)$ .

$\frac{(a - 5) (3 a - 1)}{(2 a - 3) (a - 3) (1)} \cdot \frac{(2 a - 3) (a - 3)}{(a - 2) (a + 5)}$

Step 9.2

Cancel the common factor.

$\frac{(a - 5) (3 a - 1)}{(2 a - 3) (a - 3) \cdot 1} \cdot \frac{(2 a - 3) (a - 3)}{(a - 2) (a + 5)}$

Step 9.3

Rewrite the expression.

$(a - 5) (3 a - 1) \frac{1}{(a - 2) (a + 5)}$

Step 10

Expand $(a - 5) (3 a - 1)$ using the FOIL Method.

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Step 10.1

Apply the distributive property.

$(a (3 a - 1) - 5 (3 a - 1)) \frac{1}{(a - 2) (a + 5)}$

Step 10.2

Apply the distributive property.

$(a (3 a) + a \cdot - 1 - 5 (3 a - 1)) \frac{1}{(a - 2) (a + 5)}$

Step 10.3

Apply the distributive property.

$(a (3 a) + a \cdot - 1 - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Rewrite using the commutative property of multiplication.

$(3 a \cdot a + a \cdot - 1 - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.2

Multiply $a$ by $a$ by adding the exponents.

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Step 11.1.2.1

Move $a$ .

$(3 (a \cdot a) + a \cdot - 1 - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.2.2

Multiply $a$ by $a$ .

$(3 a^{2} + a \cdot - 1 - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.3

Move $- 1$ to the left of $a$ .

$(3 a^{2} - 1 \cdot a - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.4

Rewrite $- 1 a$ as $- a$ .

$(3 a^{2} - a - 5 (3 a) - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.5

Multiply $3$ by $- 5$ .

$(3 a^{2} - a - 15 a - 5 \cdot - 1) \frac{1}{(a - 2) (a + 5)}$

Step 11.1.6

Multiply $- 5$ by $- 1$ .

$(3 a^{2} - a - 15 a + 5) \frac{1}{(a - 2) (a + 5)}$

Step 11.2

Subtract $15 a$ from $- a$ .

$(3 a^{2} - 16 a + 5) \frac{1}{(a - 2) (a + 5)}$

Step 12

Multiply $3 a^{2} - 16 a + 5$ by $\frac{1}{(a - 2) (a + 5)}$ .

$\frac{3 a^{2} - 16 a + 5}{(a - 2) (a + 5)}$

Step 13

Factor by grouping.

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Step 13.1

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 $a \cdot c = 3 \cdot 5 = 15$ and whose sum is $b = - 16$ .

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Step 13.1.1

Factor $- 16$ out of $- 16 a$ .

$\frac{3 a^{2} - 16 (a) + 5}{(a - 2) (a + 5)}$

Step 13.1.2

Rewrite $- 16$ as $- 1$ plus $- 15$

$\frac{3 a^{2} + (- 1 - 15) a + 5}{(a - 2) (a + 5)}$

Step 13.1.3

Apply the distributive property.

$\frac{3 a^{2} - 1 a - 15 a + 5}{(a - 2) (a + 5)}$

Step 13.2

Factor out the greatest common factor from each group.

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Step 13.2.1

Group the first two terms and the last two terms.

$\frac{(3 a^{2} - 1 a) - 15 a + 5}{(a - 2) (a + 5)}$

Step 13.2.2

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

$\frac{a (3 a - 1) - 5 (3 a - 1)}{(a - 2) (a + 5)}$

Step 13.3

Factor the polynomial by factoring out the greatest common factor, $3 a - 1$ .

$\frac{(3 a - 1) (a - 5)}{(a - 2) (a + 5)}$