Algebra Examples

$\frac{x + 2}{2 x^{2} + 13 x + 20} - \frac{x + 3}{2 x^{2} + 13 x + 20}$

Step 1

Combine the numerators over the common denominator.

$\frac{x + 2 - (x + 3)}{2 x^{2} + 13 x + 20}$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$\frac{x + 2 - x - 1 \cdot 3}{2 x^{2} + 13 x + 20}$

Step 2.2

Multiply $- 1$ by $3$ .

$\frac{x + 2 - x - 3}{2 x^{2} + 13 x + 20}$

Step 3

Simplify by adding terms.

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Step 3.1

Combine the opposite terms in $x + 2 - x - 3$ .

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Step 3.1.1

Subtract $x$ from $x$ .

$\frac{0 + 2 - 3}{2 x^{2} + 13 x + 20}$

Step 3.1.2

Add $0$ and $2$ .

$\frac{2 - 3}{2 x^{2} + 13 x + 20}$

Step 3.2

Subtract $3$ from $2$ .

$\frac{- 1}{2 x^{2} + 13 x + 20}$

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 $a \cdot c = 2 \cdot 20 = 40$ and whose sum is $b = 13$ .

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Step 4.1.1

Factor $13$ out of $13 x$ .

$\frac{- 1}{2 x^{2} + 13 (x) + 20}$

Step 4.1.2

Rewrite $13$ as $5$ plus $8$

$\frac{- 1}{2 x^{2} + (5 + 8) x + 20}$

Step 4.1.3

Apply the distributive property.

$\frac{- 1}{2 x^{2} + 5 x + 8 x + 20}$

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.

$\frac{- 1}{(2 x^{2} + 5 x) + 8 x + 20}$

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 5$ .

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

Step 5

Move the negative in front of the fraction.

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