Basic Math Examples

\frac{5 q + 1}{4 q^{2} + 7 q - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}

$\frac{5 q + 1}{4 q^{2} + 7 q - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1

Simplify each term.

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Step 1.1

Factor by grouping.

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Step 1.1.1

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

$a \cdot c = 4 \cdot - 2 = - 8$ and whose sum is

$b = 7$ .

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Step 1.1.1.1

Factor

$7$ out of

$7 q$ .

$\frac{5 q + 1}{4 q^{2} + 7 (q) - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.1.1.2

Rewrite

$7$ as

$- 1$ plus

$8$

$\frac{5 q + 1}{4 q^{2} + (- 1 + 8) q - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.1.1.3

Apply the distributive property.

$\frac{5 q + 1}{4 q^{2} - 1 q + 8 q - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.1.2

Factor out the greatest common factor from each group.

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Step 1.1.2.1

Group the first two terms and the last two terms.

$\frac{5 q + 1}{(4 q^{2} - 1 q) + 8 q - 2} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.1.2.2

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

$\frac{5 q + 1}{q (4 q - 1) + 2 (4 q - 1)} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.1.3

Factor the polynomial by factoring out the greatest common factor,

$4 q - 1$ .

$\frac{5 q + 1}{(4 q - 1) (q + 2)} - \frac{2 q + 3}{3 q^{2} + 5 q - 2}$

Step 1.2

Factor by grouping.

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Step 1.2.1

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

$a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is

$b = 5$ .

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Step 1.2.1.1

Factor

$5$ out of

$5 q$ .

$\frac{5 q + 1}{(4 q - 1) (q + 2)} - \frac{2 q + 3}{3 q^{2} + 5 (q) - 2}$

Step 1.2.1.2

Rewrite

$5$ as

$- 1$ plus

$6$

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

Step 1.2.1.3

Apply the distributive property.

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

Step 1.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2

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

$\frac{5 q + 1}{(4 q - 1) (q + 2)} - \frac{2 q + 3}{q (3 q - 1) + 2 (3 q - 1)}$

Step 1.2.3

Factor the polynomial by factoring out the greatest common factor,

$3 q - 1$ .

$\frac{5 q + 1}{(4 q - 1) (q + 2)} - \frac{2 q + 3}{(3 q - 1) (q + 2)}$

Step 2

To write

$\frac{5 q + 1}{(4 q - 1) (q + 2)}$ as a fraction with a common denominator, multiply by

$\frac{3 q - 1}{3 q - 1}$ .

$\frac{5 q + 1}{(4 q - 1) (q + 2)} \cdot \frac{3 q - 1}{3 q - 1} - \frac{2 q + 3}{(3 q - 1) (q + 2)}$

Step 3

To write

$- \frac{2 q + 3}{(3 q - 1) (q + 2)}$ as a fraction with a common denominator, multiply by

$\frac{4 q - 1}{4 q - 1}$ .

$\frac{5 q + 1}{(4 q - 1) (q + 2)} \cdot \frac{3 q - 1}{3 q - 1} - \frac{2 q + 3}{(3 q - 1) (q + 2)} \cdot \frac{4 q - 1}{4 q - 1}$

Step 4

Write each expression with a common denominator of

$(4 q - 1) (q + 2) (3 q - 1)$ , by multiplying each by an appropriate factor of

$1$ .

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Step 4.1

Multiply

$\frac{5 q + 1}{(4 q - 1) (q + 2)}$ by

$\frac{3 q - 1}{3 q - 1}$ .

$\frac{(5 q + 1) (3 q - 1)}{(4 q - 1) (q + 2) (3 q - 1)} - \frac{2 q + 3}{(3 q - 1) (q + 2)} \cdot \frac{4 q - 1}{4 q - 1}$

Step 4.2

Multiply

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

$\frac{4 q - 1}{4 q - 1}$ .

$\frac{(5 q + 1) (3 q - 1)}{(4 q - 1) (q + 2) (3 q - 1)} - \frac{(2 q + 3) (4 q - 1)}{(3 q - 1) (q + 2) (4 q - 1)}$

Step 4.3

Reorder the factors of

$(4 q - 1) (q + 2) (3 q - 1)$ .

$\frac{(5 q + 1) (3 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)} - \frac{(2 q + 3) (4 q - 1)}{(3 q - 1) (q + 2) (4 q - 1)}$

Step 4.4

Reorder the factors of

$(3 q - 1) (q + 2) (4 q - 1)$ .

$\frac{(5 q + 1) (3 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)} - \frac{(2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 5

Combine the numerators over the common denominator.

$\frac{(5 q + 1) (3 q - 1) - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Expand

$(5 q + 1) (3 q - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{5 q (3 q - 1) + 1 (3 q - 1) - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.1.2

Apply the distributive property.

$\frac{5 q (3 q) + 5 q \cdot - 1 + 1 (3 q - 1) - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.1.3

Apply the distributive property.

$\frac{5 q (3 q) + 5 q \cdot - 1 + 1 (3 q) + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{5 \cdot 3 q \cdot q + 5 q \cdot - 1 + 1 (3 q) + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.1.2

Multiply

$q$ by

$q$ by adding the exponents.

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Step 6.2.1.2.1

Move

$q$ .

$\frac{5 \cdot 3 (q \cdot q) + 5 q \cdot - 1 + 1 (3 q) + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.1.2.2

Multiply

$q$ by

$q$ .

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

Step 6.2.1.3

Multiply

$5$ by

$3$ .

$\frac{15 q^{2} + 5 q \cdot - 1 + 1 (3 q) + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.1.4

Multiply

$- 1$ by

$5$ .

$\frac{15 q^{2} - 5 q + 1 (3 q) + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.1.5

Multiply

$3 q$ by

$1$ .

$\frac{15 q^{2} - 5 q + 3 q + 1 \cdot - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.1.6

Multiply

$- 1$ by

$1$ .

$\frac{15 q^{2} - 5 q + 3 q - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.2.2

Add

$- 5 q$ and

$3 q$ .

$\frac{15 q^{2} - 2 q - 1 - (2 q + 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.3

Apply the distributive property.

$\frac{15 q^{2} - 2 q - 1 + (- (2 q) - 1 \cdot 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.4

Multiply

$2$ by

$- 1$ .

$\frac{15 q^{2} - 2 q - 1 + (- 2 q - 1 \cdot 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.5

Multiply

$- 1$ by

$3$ .

$\frac{15 q^{2} - 2 q - 1 + (- 2 q - 3) (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.6

Expand

$(- 2 q - 3) (4 q - 1)$ using the FOIL Method.

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Step 6.6.1

Apply the distributive property.

$\frac{15 q^{2} - 2 q - 1 - 2 q (4 q - 1) - 3 (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.6.2

Apply the distributive property.

$\frac{15 q^{2} - 2 q - 1 - 2 q (4 q) - 2 q \cdot - 1 - 3 (4 q - 1)}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.6.3

Apply the distributive property.

$\frac{15 q^{2} - 2 q - 1 - 2 q (4 q) - 2 q \cdot - 1 - 3 (4 q) - 3 \cdot - 1}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7

Simplify and combine like terms.

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Step 6.7.1

Simplify each term.

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Step 6.7.1.1

Rewrite using the commutative property of multiplication.

$\frac{15 q^{2} - 2 q - 1 - 2 \cdot 4 q \cdot q - 2 q \cdot - 1 - 3 (4 q) - 3 \cdot - 1}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7.1.2

Multiply

$q$ by

$q$ by adding the exponents.

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Step 6.7.1.2.1

Move

$q$ .

$\frac{15 q^{2} - 2 q - 1 - 2 \cdot 4 (q \cdot q) - 2 q \cdot - 1 - 3 (4 q) - 3 \cdot - 1}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7.1.2.2

Multiply

$q$ by

$q$ .

$\frac{15 q^{2} - 2 q - 1 - 2 \cdot 4 q^{2} - 2 q \cdot - 1 - 3 (4 q) - 3 \cdot - 1}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7.1.3

Multiply

$- 2$ by

$4$ .

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

Step 6.7.1.4

Multiply

$- 1$ by

$- 2$ .

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

Step 6.7.1.5

Multiply

$4$ by

$- 3$ .

$\frac{15 q^{2} - 2 q - 1 - 8 q^{2} + 2 q - 12 q - 3 \cdot - 1}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7.1.6

Multiply

$- 3$ by

$- 1$ .

$\frac{15 q^{2} - 2 q - 1 - 8 q^{2} + 2 q - 12 q + 3}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.7.2

Subtract

$12 q$ from

$2 q$ .

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

Step 6.8

Subtract

$8 q^{2}$ from

$15 q^{2}$ .

$\frac{7 q^{2} - 2 q - 1 - 10 q + 3}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.9

Subtract

$10 q$ from

$- 2 q$ .

$\frac{7 q^{2} - 12 q - 1 + 3}{(3 q - 1) (4 q - 1) (q + 2)}$

Step 6.10

Add

$- 1$ and

$3$ .

$\frac{7 q^{2} - 12 q + 2}{(3 q - 1) (4 q - 1) (q + 2)}$