Algebra Examples

$\frac{6}{m^{2} - 14 m + 45} - \frac{m + 7}{m^{2} + 2 m - 35}$

Step 1

Simplify each term.

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Step 1.1

Factor $m^{2} - 14 m + 45$ 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 $45$ and whose sum is $- 14$ .

$- 9, - 5$

Step 1.1.2

Write the factored form using these integers.

$\frac{6}{(m - 9) (m - 5)} - \frac{m + 7}{m^{2} + 2 m - 35}$

Step 1.2

Factor $m^{2} + 2 m - 35$ 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 1.2.2

Write the factored form using these integers.

$\frac{6}{(m - 9) (m - 5)} - \frac{m + 7}{(m - 5) (m + 7)}$

Step 1.3

Cancel the common factor of $m + 7$ .

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Step 1.3.1

Cancel the common factor.

$\frac{6}{(m - 9) (m - 5)} - \frac{m + 7}{(m - 5) (m + 7)}$

Step 1.3.2

Rewrite the expression.

$\frac{6}{(m - 9) (m - 5)} - \frac{1}{m - 5}$

Step 2

To write $- \frac{1}{m - 5}$ as a fraction with a common denominator, multiply by $\frac{m - 9}{m - 9}$ .

$\frac{6}{(m - 9) (m - 5)} - \frac{1}{m - 5} \cdot \frac{m - 9}{m - 9}$

Step 3

Write each expression with a common denominator of $(m - 9) (m - 5)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{m - 5}$ by $\frac{m - 9}{m - 9}$ .

$\frac{6}{(m - 9) (m - 5)} - \frac{m - 9}{(m - 5) (m - 9)}$

Step 3.2

Reorder the factors of $(m - 9) (m - 5)$ .

$\frac{6}{(m - 5) (m - 9)} - \frac{m - 9}{(m - 5) (m - 9)}$

Step 4

Combine the numerators over the common denominator.

$\frac{6 - (m - 9)}{(m - 5) (m - 9)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{6 - m - - 9}{(m - 5) (m - 9)}$

Step 5.2

Multiply $- 1$ by $- 9$ .

$\frac{6 - m + 9}{(m - 5) (m - 9)}$

Step 5.3

Add $6$ and $9$ .

$\frac{- m + 15}{(m - 5) (m - 9)}$

Step 6

Simplify with factoring out.

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Step 6.1

Factor $- 1$ out of $- m$ .

$\frac{- (m) + 15}{(m - 5) (m - 9)}$

Step 6.2

Rewrite $15$ as $- 1 (- 15)$ .

$\frac{- (m) - 1 (- 15)}{(m - 5) (m - 9)}$

Step 6.3

Factor $- 1$ out of $- (m) - 1 (- 15)$ .

$\frac{- (m - 15)}{(m - 5) (m - 9)}$

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite $- (m - 15)$ as $- 1 (m - 15)$ .

$\frac{- 1 (m - 15)}{(m - 5) (m - 9)}$

Step 6.4.2

Move the negative in front of the fraction.

$- \frac{m - 15}{(m - 5) (m - 9)}$