Basic Math Examples

$\frac{1}{a^{2} + 7 a + 10} + \frac{1}{a^{2} + 10 a + 16}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} + 7 a + 10$ 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 $10$ and whose sum is $7$ .

$2, 5$

Step 1.1.2

Write the factored form using these integers.

$\frac{1}{(a + 2) (a + 5)} + \frac{1}{a^{2} + 10 a + 16}$

Step 1.2

Factor $a^{2} + 10 a + 16$ 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 $16$ and whose sum is $10$ .

$2, 8$

Step 1.2.2

Write the factored form using these integers.

$\frac{1}{(a + 2) (a + 5)} + \frac{1}{(a + 2) (a + 8)}$

Step 2

To write $\frac{1}{(a + 2) (a + 5)}$ as a fraction with a common denominator, multiply by $\frac{a + 8}{a + 8}$ .

$\frac{1}{(a + 2) (a + 5)} \cdot \frac{a + 8}{a + 8} + \frac{1}{(a + 2) (a + 8)}$

Step 3

To write $\frac{1}{(a + 2) (a + 8)}$ as a fraction with a common denominator, multiply by $\frac{a + 5}{a + 5}$ .

$\frac{1}{(a + 2) (a + 5)} \cdot \frac{a + 8}{a + 8} + \frac{1}{(a + 2) (a + 8)} \cdot \frac{a + 5}{a + 5}$

Step 4

Write each expression with a common denominator of $(a + 2) (a + 5) (a + 8)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{1}{(a + 2) (a + 5)}$ by $\frac{a + 8}{a + 8}$ .

$\frac{a + 8}{(a + 2) (a + 5) (a + 8)} + \frac{1}{(a + 2) (a + 8)} \cdot \frac{a + 5}{a + 5}$

Step 4.2

Multiply $\frac{1}{(a + 2) (a + 8)}$ by $\frac{a + 5}{a + 5}$ .

$\frac{a + 8}{(a + 2) (a + 5) (a + 8)} + \frac{a + 5}{(a + 2) (a + 8) (a + 5)}$

Step 4.3

Reorder the factors of $(a + 2) (a + 8) (a + 5)$ .

$\frac{a + 8}{(a + 2) (a + 5) (a + 8)} + \frac{a + 5}{(a + 2) (a + 5) (a + 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{a + 8 + a + 5}{(a + 2) (a + 5) (a + 8)}$

Step 6

Simplify the numerator.

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Step 6.1

Add $a$ and $a$ .

$\frac{2 a + 8 + 5}{(a + 2) (a + 5) (a + 8)}$

Step 6.2

Add $8$ and $5$ .

$\frac{2 a + 13}{(a + 2) (a + 5) (a + 8)}$