Basic Math Examples

$\frac{4}{a^{2} - 8 a + 15} - \frac{3}{a^{2} - 10 a + 25}$

Step 1

Simplify each term.

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Step 1.1

Factor $a^{2} - 8 a + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 1.1.2

Write the factored form using these integers.

$\frac{4}{(a - 5) (a - 3)} - \frac{3}{a^{2} - 10 a + 25}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

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

$\frac{4}{(a - 5) (a - 3)} - \frac{3}{a^{2} - 10 a + 5^{2}}$

Step 1.2.2

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 1.2.3

Rewrite the polynomial.

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

Step 1.2.4

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

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

Step 2

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

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

Step 3

To write $- \frac{3}{{(a - 5)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{a - 3}{a - 3}$ .

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

Step 4

Write each expression with a common denominator of $(a - 3) {(a - 5)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Add $1$ and $1$ .

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

Step 4.6

Multiply $\frac{3}{{(a - 5)}^{2}}$ by $\frac{a - 3}{a - 3}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $4$ by $- 5$ .

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

Step 6.3

Apply the distributive property.

$\frac{4 a - 20 - 3 a - 3 \cdot - 3}{{(a - 5)}^{2} (a - 3)}$

Step 6.4

Multiply $- 3$ by $- 3$ .

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

Step 6.5

Subtract $3 a$ from $4 a$ .

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

Step 6.6

Add $- 20$ and $9$ .

$\frac{a - 11}{{(a - 5)}^{2} (a - 3)}$