Algebra Examples

$\frac{2 x^{2} - 3 x - 5}{25 - 4 x^{2}}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 3$ .

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Step 1.1.1

Factor $- 3$ out of $- 3 x$ .

$\frac{2 x^{2} - 3 (x) - 5}{25 - 4 x^{2}}$

Step 1.1.2

Rewrite $- 3$ as $2$ plus $- 5$

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

Step 1.1.3

Apply the distributive property.

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

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

$\frac{2 x (x + 1) - 5 (x + 1)}{25 - 4 x^{2}}$

Step 1.3

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

$\frac{(x + 1) (2 x - 5)}{25 - 4 x^{2}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{(x + 1) (2 x - 5)}{5^{2} - 4 x^{2}}$

Step 2.2

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

$\frac{(x + 1) (2 x - 5)}{5^{2} - {(2 x)}^{2}}$

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

$\frac{(x + 1) (2 x - 5)}{(5 + 2 x) (5 - (2 x))}$

Step 2.4

Multiply $2$ by $- 1$ .

$\frac{(x + 1) (2 x - 5)}{(5 + 2 x) (5 - 2 x)}$

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $2 x - 5$ and $5 - 2 x$ .

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Step 3.1.1

Factor $- 1$ out of $2 x$ .

$\frac{(x + 1) (- (- 2 x) - 5)}{(5 + 2 x) (5 - 2 x)}$

Step 3.1.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{(x + 1) (- (- 2 x) - 1 (5))}{(5 + 2 x) (5 - 2 x)}$

Step 3.1.3

Factor $- 1$ out of $- (- 2 x) - 1 (5)$ .

$\frac{(x + 1) (- (- 2 x + 5))}{(5 + 2 x) (5 - 2 x)}$

Step 3.1.4

Rewrite $- (- 2 x + 5)$ as $- 1 (- 2 x + 5)$ .

$\frac{(x + 1) (- 1 (- 2 x + 5))}{(5 + 2 x) (5 - 2 x)}$

Step 3.1.5

Reorder terms.

$\frac{(x + 1) (- 1 (- 2 x + 5))}{(5 + 2 x) (- 2 x + 5)}$

Step 3.1.6

Cancel the common factor.

$\frac{(x + 1) (- 1 (- 2 x + 5))}{(5 + 2 x) (- 2 x + 5)}$

Step 3.1.7

Rewrite the expression.

$\frac{(x + 1) \cdot (- 1)}{5 + 2 x}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $- 1$ to the left of $x + 1$ .

$\frac{- 1 \cdot (x + 1)}{5 + 2 x}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{x + 1}{5 + 2 x}$