Solve a+2/a^2-5a+6+a+2/a^2-7a+12+1/a^2-6a+8

$$\frac { a + 2 } { a ^ { 2 } - 5 a + 6 } + \frac { a + 2 } { a ^ { 2 } - 7 a + 12 } + \frac { 1 } { a ^ { 2 } - 6 a + 8 }$$

Evaluate

$$\frac{2a+5}{\left(a-4\right)\left(a-2\right)}$$

Short Solution Steps

Factor a^{2}-5a+6. Factor a^{2}-7a+12.

$$\frac{a+2}{\left(a-3\right)\left(a-2\right)}+\frac{a+2}{\left(a-4\right)\left(a-3\right)}+\frac{1}{a^{2}-6a+8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(a-2\right) and \left(a-4\right)\left(a-3\right) is \left(a-4\right)\left(a-3\right)\left(a-2\right). Multiply \frac{a+2}{\left(a-3\right)\left(a-2\right)} times \frac{a-4}{a-4}. Multiply \frac{a+2}{\left(a-4\right)\left(a-3\right)} times \frac{a-2}{a-2}.

$$\frac{\left(a+2\right)\left(a-4\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Since \frac{\left(a+2\right)\left(a-4\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)} and \frac{\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+2\right)\left(a-4\right)+\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Do the multiplications in \left(a+2\right)\left(a-4\right)+\left(a+2\right)\left(a-2\right).

$$\frac{a^{2}-4a+2a-8+a^{2}-2a+2a-4}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Combine like terms in a^{2}-4a+2a-8+a^{2}-2a+2a-4.

$$\frac{2a^{2}-2a-12}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Factor the expressions that are not already factored in \frac{2a^{2}-2a-12}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}.

$$\frac{2\left(a-3\right)\left(a+2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Cancel out a-3 in both numerator and denominator.

$$\frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Factor a^{2}-6a+8.

$$\frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)}+\frac{1}{\left(a-4\right)\left(a-2\right)}$$

Since \frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)} and \frac{1}{\left(a-4\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a+2\right)+1}{\left(a-4\right)\left(a-2\right)}$$

Do the multiplications in 2\left(a+2\right)+1.

$$\frac{2a+4+1}{\left(a-4\right)\left(a-2\right)}$$

Combine like terms in 2a+4+1.

$$\frac{2a+5}{\left(a-4\right)\left(a-2\right)}$$

Expand \left(a-4\right)\left(a-2\right).

$$\frac{2a+5}{a^{2}-6a+8}$$

Expand

$$\frac{2a+5}{\left(a-4\right)\left(a-2\right)}$$

Short Solution Steps

Factor a^{2}-5a+6. Factor a^{2}-7a+12.

$$\frac{a+2}{\left(a-3\right)\left(a-2\right)}+\frac{a+2}{\left(a-4\right)\left(a-3\right)}+\frac{1}{a^{2}-6a+8}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-3\right)\left(a-2\right) and \left(a-4\right)\left(a-3\right) is \left(a-4\right)\left(a-3\right)\left(a-2\right). Multiply \frac{a+2}{\left(a-3\right)\left(a-2\right)} times \frac{a-4}{a-4}. Multiply \frac{a+2}{\left(a-4\right)\left(a-3\right)} times \frac{a-2}{a-2}.

$$\frac{\left(a+2\right)\left(a-4\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Since \frac{\left(a+2\right)\left(a-4\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)} and \frac{\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+2\right)\left(a-4\right)+\left(a+2\right)\left(a-2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Do the multiplications in \left(a+2\right)\left(a-4\right)+\left(a+2\right)\left(a-2\right).

$$\frac{a^{2}-4a+2a-8+a^{2}-2a+2a-4}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Combine like terms in a^{2}-4a+2a-8+a^{2}-2a+2a-4.

$$\frac{2a^{2}-2a-12}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Factor the expressions that are not already factored in \frac{2a^{2}-2a-12}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}.

$$\frac{2\left(a-3\right)\left(a+2\right)}{\left(a-4\right)\left(a-3\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Cancel out a-3 in both numerator and denominator.

$$\frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)}+\frac{1}{a^{2}-6a+8}$$

Factor a^{2}-6a+8.

$$\frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)}+\frac{1}{\left(a-4\right)\left(a-2\right)}$$

Since \frac{2\left(a+2\right)}{\left(a-4\right)\left(a-2\right)} and \frac{1}{\left(a-4\right)\left(a-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{2\left(a+2\right)+1}{\left(a-4\right)\left(a-2\right)}$$

Do the multiplications in 2\left(a+2\right)+1.

$$\frac{2a+4+1}{\left(a-4\right)\left(a-2\right)}$$

Combine like terms in 2a+4+1.

$$\frac{2a+5}{\left(a-4\right)\left(a-2\right)}$$

Expand \left(a-4\right)\left(a-2\right).

$$\frac{2a+5}{a^{2}-6a+8}$$