Basic Math Examples

$\frac{3 a}{a^{2} - 4 a - 5} + \frac{4}{a^{2} - 9 a + 20}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} - 4 a - 5$ using the AC method.

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Step 1.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 1.1.2

Write the factored form using these integers.

$\frac{3 a}{(a - 5) (a + 1)} + \frac{4}{a^{2} - 9 a + 20}$

Step 1.2

Factor $a^{2} - 9 a + 20$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $- 9$ .

$- 5, - 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{3 a}{(a - 5) (a + 1)} + \frac{4}{(a - 5) (a - 4)}$

Step 2

To write $\frac{3 a}{(a - 5) (a + 1)}$ as a fraction with a common denominator, multiply by $\frac{a - 4}{a - 4}$ .

$\frac{3 a}{(a - 5) (a + 1)} \cdot \frac{a - 4}{a - 4} + \frac{4}{(a - 5) (a - 4)}$

Step 3

To write $\frac{4}{(a - 5) (a - 4)}$ as a fraction with a common denominator, multiply by $\frac{a + 1}{a + 1}$ .

$\frac{3 a}{(a - 5) (a + 1)} \cdot \frac{a - 4}{a - 4} + \frac{4}{(a - 5) (a - 4)} \cdot \frac{a + 1}{a + 1}$

Step 4

Write each expression with a common denominator of $(a - 5) (a + 1) (a - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{3 a}{(a - 5) (a + 1)}$ by $\frac{a - 4}{a - 4}$ .

$\frac{3 a (a - 4)}{(a - 5) (a + 1) (a - 4)} + \frac{4}{(a - 5) (a - 4)} \cdot \frac{a + 1}{a + 1}$

Step 4.2

Multiply $\frac{4}{(a - 5) (a - 4)}$ by $\frac{a + 1}{a + 1}$ .

$\frac{3 a (a - 4)}{(a - 5) (a + 1) (a - 4)} + \frac{4 (a + 1)}{(a - 5) (a - 4) (a + 1)}$

Step 4.3

Reorder the factors of $(a - 5) (a + 1) (a - 4)$ .

$\frac{3 a (a - 4)}{(a + 1) (a - 4) (a - 5)} + \frac{4 (a + 1)}{(a - 5) (a - 4) (a + 1)}$

Step 4.4

Reorder the factors of $(a - 5) (a - 4) (a + 1)$ .

$\frac{3 a (a - 4)}{(a + 1) (a - 4) (a - 5)} + \frac{4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{3 a (a - 4) + 4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{3 a \cdot a + 3 a \cdot - 4 + 4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 6.2

Multiply $a$ by $a$ by adding the exponents.

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Step 6.2.1

Move $a$ .

$\frac{3 (a \cdot a) + 3 a \cdot - 4 + 4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 6.2.2

Multiply $a$ by $a$ .

$\frac{3 a^{2} + 3 a \cdot - 4 + 4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 6.3

Multiply $- 4$ by $3$ .

$\frac{3 a^{2} - 12 a + 4 (a + 1)}{(a + 1) (a - 4) (a - 5)}$

Step 6.4

Apply the distributive property.

$\frac{3 a^{2} - 12 a + 4 a + 4 \cdot 1}{(a + 1) (a - 4) (a - 5)}$

Step 6.5

Multiply $4$ by $1$ .

$\frac{3 a^{2} - 12 a + 4 a + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.6

Add $- 12 a$ and $4 a$ .

$\frac{3 a^{2} - 8 a + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.7

Factor by grouping.

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Step 6.7.1

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = - 8$ .

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Step 6.7.1.1

Factor $- 8$ out of $- 8 a$ .

$\frac{3 a^{2} - 8 (a) + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.7.1.2

Rewrite $- 8$ as $- 2$ plus $- 6$

$\frac{3 a^{2} + (- 2 - 6) a + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.7.1.3

Apply the distributive property.

$\frac{3 a^{2} - 2 a - 6 a + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.7.2

Factor out the greatest common factor from each group.

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Step 6.7.2.1

Group the first two terms and the last two terms.

$\frac{(3 a^{2} - 2 a) - 6 a + 4}{(a + 1) (a - 4) (a - 5)}$

Step 6.7.2.2

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

$\frac{a (3 a - 2) - 2 (3 a - 2)}{(a + 1) (a - 4) (a - 5)}$

Step 6.7.3

Factor the polynomial by factoring out the greatest common factor, $3 a - 2$ .

$\frac{(3 a - 2) (a - 2)}{(a + 1) (a - 4) (a - 5)}$