Simplify (-5z^2+9z)/(4z^2-25)-(-7z^2+4z+25)/(4z^2-25)

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

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.

Apply the distributive property.

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

Simplify.

Multiply $- 7$ by $- 1$ .

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

Multiply $4$ by $- 1$ .

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

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.

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

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

Subtract $4 z$ from $9 z$ .

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

Step 4

Factor by grouping.

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 $b = 5$ .

Factor $5$ out of $5 z$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

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

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.

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

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

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

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

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 $2 z - 5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (-5z^2+9z)/(4z^2-25)-(-7z^2+4z+25)/(4z^2-25)

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