Algebra Examples

$\frac{- 5 z^{2} + 9 z}{4 z^{2} - 25} - \frac{- 7 z^{2} + 4 z + 25}{4 z^{2} - 25}$

Step 1

Combine the numerators over the common denominator.

$\frac{- 5 z^{2} + 9 z - (- 7 z^{2} + 4 z + 25)}{4 z^{2} - 25}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{- 5 z^{2} + 9 z - (- 7 z^{2}) - (4 z) - 1 \cdot 25}{4 z^{2} - 25}$

Step 2.2

Simplify.

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Step 2.2.1

Multiply $- 7$ by $- 1$ .

$\frac{- 5 z^{2} + 9 z + 7 z^{2} - (4 z) - 1 \cdot 25}{4 z^{2} - 25}$

Step 2.2.2

Multiply $4$ by $- 1$ .

$\frac{- 5 z^{2} + 9 z + 7 z^{2} - 4 z - 1 \cdot 25}{4 z^{2} - 25}$

Step 2.2.3

Multiply $- 1$ by $25$ .

$\frac{- 5 z^{2} + 9 z + 7 z^{2} - 4 z - 25}{4 z^{2} - 25}$

Step 3

Simplify by adding terms.

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Step 3.1

Add $- 5 z^{2}$ and $7 z^{2}$ .

$\frac{2 z^{2} + 9 z - 4 z - 25}{4 z^{2} - 25}$

Step 3.2

Subtract $4 z$ from $9 z$ .

$\frac{2 z^{2} + 5 z - 25}{4 z^{2} - 25}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 2 \cdot - 25 = - 50$ and whose sum is $b = 5$ .

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Step 4.1.1

Factor $5$ out of $5 z$ .

$\frac{2 z^{2} + 5 (z) - 25}{4 z^{2} - 25}$

Step 4.1.2

Rewrite $5$ as $- 5$ plus $10$

$\frac{2 z^{2} + (- 5 + 10) z - 25}{4 z^{2} - 25}$

Step 4.1.3

Apply the distributive property.

$\frac{2 z^{2} - 5 z + 10 z - 25}{4 z^{2} - 25}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(2 z^{2} - 5 z) + 10 z - 25}{4 z^{2} - 25}$

Step 4.2.2

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

$\frac{z (2 z - 5) + 5 (2 z - 5)}{4 z^{2} - 25}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 z - 5$ .

$\frac{(2 z - 5) (z + 5)}{4 z^{2} - 25}$

Step 5

Simplify the denominator.

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Step 5.1

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

$\frac{(2 z - 5) (z + 5)}{{(2 z)}^{2} - 25}$

Step 5.2

Rewrite $25$ as $5^{2}$ .

$\frac{(2 z - 5) (z + 5)}{{(2 z)}^{2} - 5^{2}}$

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 $a = 2 z$ and $b = 5$ .

$\frac{(2 z - 5) (z + 5)}{(2 z + 5) (2 z - 5)}$

Step 6

Cancel the common factor of $2 z - 5$ .

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Step 6.1

Cancel the common factor.

$\frac{(2 z - 5) (z + 5)}{(2 z + 5) (2 z - 5)}$

Step 6.2

Rewrite the expression.

$\frac{z + 5}{2 z + 5}$