Basic Math Examples

$\frac{4 z^{2} - 9}{2 z^{2} - z - 3}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(2 z)}^{2} - 9}{2 z^{2} - z - 3}$

Step 1.2

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

$\frac{{(2 z)}^{2} - 3^{2}}{2 z^{2} - z - 3}$

Step 1.3

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

$\frac{(2 z + 3) (2 z - 3)}{2 z^{2} - z - 3}$

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

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Step 2.1.1

Factor $- 1$ out of $- z$ .

$\frac{(2 z + 3) (2 z - 3)}{2 z^{2} - (z) - 3}$

Step 2.1.2

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

$\frac{(2 z + 3) (2 z - 3)}{2 z^{2} + (2 - 3) z - 3}$

Step 2.1.3

Apply the distributive property.

$\frac{(2 z + 3) (2 z - 3)}{2 z^{2} + 2 z - 3 z - 3}$

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.

$\frac{(2 z + 3) (2 z - 3)}{(2 z^{2} + 2 z) - 3 z - 3}$

Step 2.2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

$\frac{2 z + 3}{z + 1}$