Solve 5a/a-9+a^2-2a+1/a^2-12a+27-6a/(a-3)

$$\frac { 5 a } { a - 9 } + \frac { a ^ { 2 } - 2 a + 1 } { a ^ { 2 } - 12 a + 27 } - \frac { 6 a } { ( a - 3 ) }$$

Evaluate

$$\frac{37a+1}{\left(a-9\right)\left(a-3\right)}$$

Short Solution Steps

Factor a^{2}-12a+27.

$$\frac{5a}{a-9}+\frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-9 and \left(a-9\right)\left(a-3\right) is \left(a-9\right)\left(a-3\right). Multiply \frac{5a}{a-9} times \frac{a-3}{a-3}.

$$\frac{5a\left(a-3\right)}{\left(a-9\right)\left(a-3\right)}+\frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Since \frac{5a\left(a-3\right)}{\left(a-9\right)\left(a-3\right)} and \frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5a\left(a-3\right)+a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Do the multiplications in 5a\left(a-3\right)+a^{2}-2a+1.

$$\frac{5a^{2}-15a+a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Combine like terms in 5a^{2}-15a+a^{2}-2a+1.

$$\frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-9\right)\left(a-3\right) and a-3 is \left(a-9\right)\left(a-3\right). Multiply \frac{6a}{a-3} times \frac{a-9}{a-9}.

$$\frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)}$$

Since \frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)} and \frac{6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6a^{2}-17a+1-6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)}$$

Do the multiplications in 6a^{2}-17a+1-6a\left(a-9\right).

$$\frac{6a^{2}-17a+1-6a^{2}+54a}{\left(a-9\right)\left(a-3\right)}$$

Combine like terms in 6a^{2}-17a+1-6a^{2}+54a.

$$\frac{37a+1}{\left(a-9\right)\left(a-3\right)}$$

Expand \left(a-9\right)\left(a-3\right).

$$\frac{37a+1}{a^{2}-12a+27}$$

Expand

$$\frac{37a+1}{\left(a-9\right)\left(a-3\right)}$$

Short Solution Steps

Factor a^{2}-12a+27.

$$\frac{5a}{a-9}+\frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-9 and \left(a-9\right)\left(a-3\right) is \left(a-9\right)\left(a-3\right). Multiply \frac{5a}{a-9} times \frac{a-3}{a-3}.

$$\frac{5a\left(a-3\right)}{\left(a-9\right)\left(a-3\right)}+\frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Since \frac{5a\left(a-3\right)}{\left(a-9\right)\left(a-3\right)} and \frac{a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{5a\left(a-3\right)+a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Do the multiplications in 5a\left(a-3\right)+a^{2}-2a+1.

$$\frac{5a^{2}-15a+a^{2}-2a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

Combine like terms in 5a^{2}-15a+a^{2}-2a+1.

$$\frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a}{a-3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(a-9\right)\left(a-3\right) and a-3 is \left(a-9\right)\left(a-3\right). Multiply \frac{6a}{a-3} times \frac{a-9}{a-9}.

$$\frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)}-\frac{6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)}$$

Since \frac{6a^{2}-17a+1}{\left(a-9\right)\left(a-3\right)} and \frac{6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{6a^{2}-17a+1-6a\left(a-9\right)}{\left(a-9\right)\left(a-3\right)}$$

Do the multiplications in 6a^{2}-17a+1-6a\left(a-9\right).

$$\frac{6a^{2}-17a+1-6a^{2}+54a}{\left(a-9\right)\left(a-3\right)}$$

Combine like terms in 6a^{2}-17a+1-6a^{2}+54a.

$$\frac{37a+1}{\left(a-9\right)\left(a-3\right)}$$

Expand \left(a-9\right)\left(a-3\right).

$$\frac{37a+1}{a^{2}-12a+27}$$