Basic Math Examples

$\frac{4 y^{2} - 9}{2 y^{2} + 11 y - 21} \div \frac{2 y^{2} + y - 3}{y^{2} + 6 y - 7}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 y^{2} - 9}{2 y^{2} + 11 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 2

Simplify the numerator.

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Step 2.1

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

$\frac{{(2 y)}^{2} - 9}{2 y^{2} + 11 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 2.2

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

$\frac{{(2 y)}^{2} - 3^{2}}{2 y^{2} + 11 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 2.3

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

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + 11 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

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 $a \cdot c = 2 \cdot - 21 = - 42$ and whose sum is $b = 11$ .

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Step 3.1.1

Factor $11$ out of $11 y$ .

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + 11 (y) - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 3.1.2

Rewrite $11$ as $- 3$ plus $14$

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} + (- 3 + 14) y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 3.1.3

Apply the distributive property.

$\frac{(2 y + 3) (2 y - 3)}{2 y^{2} - 3 y + 14 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

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.

$\frac{(2 y + 3) (2 y - 3)}{(2 y^{2} - 3 y) + 14 y - 21} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 3.2.2

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

$\frac{(2 y + 3) (2 y - 3)}{y (2 y - 3) + 7 (2 y - 3)} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 3.3

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

$\frac{(2 y + 3) (2 y - 3)}{(2 y - 3) (y + 7)} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 4

Cancel the common factor of $2 y - 3$ .

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Step 4.1

Cancel the common factor.

$\frac{(2 y + 3) (2 y - 3)}{(2 y - 3) (y + 7)} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 4.2

Rewrite the expression.

$\frac{2 y + 3}{y + 7} \cdot \frac{y^{2} + 6 y - 7}{2 y^{2} + y - 3}$

Step 5

Factor $y^{2} + 6 y - 7$ using the AC method.

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Step 5.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 5.2

Write the factored form using these integers.

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{2 y^{2} + y - 3}$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

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Step 6.1.1

Multiply by $1$ .

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{2 y^{2} + 1 y - 3}$

Step 6.1.2

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

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{2 y^{2} + (- 2 + 3) y - 3}$

Step 6.1.3

Apply the distributive property.

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{2 y^{2} - 2 y + 3 y - 3}$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{(2 y^{2} - 2 y) + 3 y - 3}$

Step 6.2.2

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

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{2 y (y - 1) + 3 (y - 1)}$

Step 6.3

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

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{(y - 1) (2 y + 3)}$

Step 7

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

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Step 7.1

Factor $2 y + 3$ out of $(y - 1) (2 y + 3)$ .

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{(2 y + 3) (y - 1)}$

Step 7.2

Cancel the common factor.

$\frac{2 y + 3}{y + 7} \cdot \frac{(y - 1) (y + 7)}{(2 y + 3) (y - 1)}$

Step 7.3

Rewrite the expression.

$\frac{1}{y + 7} \cdot \frac{(y - 1) (y + 7)}{y - 1}$

Step 8

Cancel the common factor of $y + 7$ .

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Step 8.1

Factor $y + 7$ out of $(y - 1) (y + 7)$ .

$\frac{1}{y + 7} \cdot \frac{(y + 7) (y - 1)}{y - 1}$

Step 8.2

Cancel the common factor.

$\frac{1}{y + 7} \cdot \frac{(y + 7) (y - 1)}{y - 1}$

Step 8.3

Rewrite the expression.

$\frac{y - 1}{y - 1}$

Step 9

Cancel the common factor of $y - 1$ .

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Step 9.1

Cancel the common factor.

$\frac{y - 1}{y - 1}$

Step 9.2

Rewrite the expression.

$1$