Basic Math Examples

$\frac{m - 2}{m^{2} - 7 m + 10} + \frac{m + 2}{m^{2} - 4 m - 5}$

Step 1

Simplify each term.

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Step 1.1

Factor $m^{2} - 7 m + 10$ 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 $- 7$ .

$- 5, - 2$

Step 1.1.2

Write the factored form using these integers.

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

Step 1.2

Cancel the common factor of $m - 2$ .

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Step 1.2.1

Cancel the common factor.

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

Step 1.2.2

Rewrite the expression.

$\frac{1}{m - 5} + \frac{m + 2}{m^{2} - 4 m - 5}$

Step 1.3

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

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Step 1.3.1

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 1.3.2

Write the factored form using these integers.

$\frac{1}{m - 5} + \frac{m + 2}{(m - 5) (m + 1)}$

Step 2

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

$\frac{1}{m - 5} \cdot \frac{m + 1}{m + 1} + \frac{m + 2}{(m - 5) (m + 1)}$

Step 3

Simplify terms.

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Step 3.1

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

$\frac{m + 1}{(m - 5) (m + 1)} + \frac{m + 2}{(m - 5) (m + 1)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{m + 1 + m + 2}{(m - 5) (m + 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Add $m$ and $m$ .

$\frac{2 m + 1 + 2}{(m - 5) (m + 1)}$

Step 4.2

Add $1$ and $2$ .

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