Basic Math Examples

\frac{10 y^{2} - 3 y - 4}{5 y^{2} - y - 4} \div \frac{1 - 4 y^{2}}{15 y^{2} + 12 y}

$\frac{10 y^{2} - 3 y - 4}{5 y^{2} - y - 4} \div \frac{1 - 4 y^{2}}{15 y^{2} + 12 y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

\frac{10 y^{2} - 3 y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}

$\frac{10 y^{2} - 3 y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2

Factor by grouping.

Tap for more steps...

Step 2.1

For a polynomial of the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is

a \cdot c = 10 \cdot - 4 = - 40

$a \cdot c = 10 \cdot - 4 = - 40$ and whose sum is

b = - 3

$b = - 3$ .

Tap for more steps...

Step 2.1.1

Factor

- 3

$- 3$ out of

- 3 y

$- 3 y$ .

\frac{10 y^{2} - 3 (y) - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}

$\frac{10 y^{2} - 3 (y) - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2.1.2

Rewrite

- 3

$- 3$ as

5

$5$ plus

- 8

$- 8$

\frac{10 y^{2} + (5 - 8) y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}

$\frac{10 y^{2} + (5 - 8) y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2.1.3

Apply the distributive property.

$\frac{10 y^{2} + 5 y - 8 y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(10 y^{2} + 5 y) - 8 y - 4}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2.2.2

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

$\frac{5 y (2 y + 1) - 4 (2 y + 1)}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor,

$2 y + 1$ .

$\frac{(2 y + 1) (5 y - 4)}{5 y^{2} - y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3

Factor by grouping.

Tap for more steps...

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

$a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is

$b = - 1$ .

Tap for more steps...

Step 3.1.1

Factor

$- 1$ out of

$- y$ .

$\frac{(2 y + 1) (5 y - 4)}{5 y^{2} - (y) - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3.1.2

Rewrite

$- 1$ as

$4$ plus

$- 5$

$\frac{(2 y + 1) (5 y - 4)}{5 y^{2} + (4 - 5) y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3.1.3

Apply the distributive property.

$\frac{(2 y + 1) (5 y - 4)}{5 y^{2} + 4 y - 5 y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(2 y + 1) (5 y - 4)}{(5 y^{2} + 4 y) - 5 y - 4} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3.2.2

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

$\frac{(2 y + 1) (5 y - 4)}{y (5 y + 4) - (5 y + 4)} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor,

$5 y + 4$ .

$\frac{(2 y + 1) (5 y - 4)}{(5 y + 4) (y - 1)} \cdot \frac{15 y^{2} + 12 y}{1 - 4 y^{2}}$

Step 4

Factor

$3 y$ out of

$15 y^{2} + 12 y$ .

Tap for more steps...

Step 4.1

Factor

$3 y$ out of

$15 y^{2}$ .

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

Step 4.2

Factor

$3 y$ out of

$12 y$ .

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

Step 4.3

Factor

$3 y$ out of

$3 y (5 y) + 3 y (4)$ .

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

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Rewrite

$1$ as

$1^{2}$ .

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

Step 5.2

Rewrite

$4 y^{2}$ as

${(2 y)}^{2}$ .

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

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

$a = 1$ and

$b = 2 y$ .

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

Step 5.4

Multiply

$2$ by

$- 1$ .

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

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

Cancel the common factor of

$5 y + 4$ .

Tap for more steps...

Step 6.1.1

Factor

$5 y + 4$ out of

$3 y (5 y + 4)$ .

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

Step 6.1.2

Cancel the common factor.

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

Step 6.1.3

Rewrite the expression.

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

Step 6.2

Multiply

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

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

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

Step 6.3

Cancel the common factor of

$2 y + 1$ and

$1 + 2 y$ .

Tap for more steps...

Step 6.3.1

Reorder terms.

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

Step 6.3.2

Cancel the common factor.

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

Step 6.3.3

Rewrite the expression.

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

Step 6.4

Reorder.

Tap for more steps...

Step 6.4.1

Move

$3$ to the left of

$5 y - 4$ .

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

Step 6.4.2

Reorder factors in

$\frac{3 (5 y - 4) y}{(y - 1) (1 - 2 y)}$ .

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