Basic Math Examples

$\frac{a}{a^{2} + 10 a + 25} - \frac{4 - a}{a^{2} + 6 a + 5}$

Step 1

Simplify each term.

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Step 1.1

Factor using the perfect square rule.

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Step 1.1.1

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

$\frac{a}{a^{2} + 10 a + 5^{2}} - \frac{4 - a}{a^{2} + 6 a + 5}$

Step 1.1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$10 a = 2 \cdot a \cdot 5$

Step 1.1.3

Rewrite the polynomial.

$\frac{a}{a^{2} + 2 \cdot a \cdot 5 + 5^{2}} - \frac{4 - a}{a^{2} + 6 a + 5}$

Step 1.1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 5$ .

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

Step 1.2

Factor $a^{2} + 6 a + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Step 1.2.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Raise $a + 5$ to the power of $1$ .

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

Step 4.4

Raise $a + 5$ to the power of $1$ .

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

Step 4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

Reorder the factors of $(a + 1) {(a + 5)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $a$ by $a$ .

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

Step 6.3

Multiply $a$ by $1$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 1$ by $4$ .

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

Step 6.6

Multiply $- - a$ .

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Step 6.6.1

Multiply $- 1$ by $- 1$ .

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

Step 6.6.2

Multiply $a$ by $1$ .

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

Step 6.7

Expand $(- 4 + a) (a + 5)$ using the FOIL Method.

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Step 6.7.1

Apply the distributive property.

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

Step 6.7.2

Apply the distributive property.

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

Step 6.7.3

Apply the distributive property.

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

Step 6.8

Simplify and combine like terms.

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Step 6.8.1

Simplify each term.

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Step 6.8.1.1

Multiply $- 4$ by $5$ .

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

Step 6.8.1.2

Multiply $a$ by $a$ .

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

Step 6.8.1.3

Move $5$ to the left of $a$ .

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

Step 6.8.2

Add $- 4 a$ and $5 a$ .

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

Step 6.9

Add $a^{2}$ and $a^{2}$ .

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

Step 6.10

Add $a$ and $a$ .

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

Step 6.11

Factor $2$ out of $2 a^{2} + 2 a - 20$ .

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Step 6.11.1

Factor $2$ out of $2 a^{2}$ .

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

Step 6.11.2

Factor $2$ out of $2 a$ .

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

Step 6.11.3

Factor $2$ out of $- 20$ .

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

Step 6.11.4

Factor $2$ out of $2 (a^{2}) + 2 a$ .

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

Step 6.11.5

Factor $2$ out of $2 (a^{2} + a) + 2 \cdot - 10$ .

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