Solve (a-9+25/a+1)div(a-1-10+1/a+1)

$$( a - 9 + \frac { 25 } { a + 1 } ) \div ( a - 1 - \frac { 10 + 1 } { a + 1 } )$$

Evaluate

$$\frac{\left(a-4\right)^{2}}{a^{2}-12}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a-9 times \frac{a+1}{a+1}.

$$\frac{\frac{\left(a-9\right)\left(a+1\right)}{a+1}+\frac{25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Since \frac{\left(a-9\right)\left(a+1\right)}{a+1} and \frac{25}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(a-9\right)\left(a+1\right)+25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Do the multiplications in \left(a-9\right)\left(a+1\right)+25.

$$\frac{\frac{a^{2}+a-9a-9+25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Combine like terms in a^{2}+a-9a-9+25.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Add 10 and 1 to get 11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{a-1-\frac{11}{a+1}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a+1}{a+1}.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{\left(a-1\right)\left(a+1\right)}{a+1}-\frac{11}{a+1}}$$

Since \frac{\left(a-1\right)\left(a+1\right)}{a+1} and \frac{11}{a+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{\left(a-1\right)\left(a+1\right)-11}{a+1}}$$

Do the multiplications in \left(a-1\right)\left(a+1\right)-11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{a^{2}+a-a-1-11}{a+1}}$$

Combine like terms in a^{2}+a-a-1-11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{a^{2}-12}{a+1}}$$

Divide \frac{a^{2}-8a+16}{a+1} by \frac{a^{2}-12}{a+1} by multiplying \frac{a^{2}-8a+16}{a+1} by the reciprocal of \frac{a^{2}-12}{a+1}.

$$\frac{\left(a^{2}-8a+16\right)\left(a+1\right)}{\left(a+1\right)\left(a^{2}-12\right)}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{a^{2}-8a+16}{a^{2}-12}$$

Expand

$$\frac{a^{2}-8a+16}{a^{2}-12}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a-9 times \frac{a+1}{a+1}.

$$\frac{\frac{\left(a-9\right)\left(a+1\right)}{a+1}+\frac{25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Since \frac{\left(a-9\right)\left(a+1\right)}{a+1} and \frac{25}{a+1} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(a-9\right)\left(a+1\right)+25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Do the multiplications in \left(a-9\right)\left(a+1\right)+25.

$$\frac{\frac{a^{2}+a-9a-9+25}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Combine like terms in a^{2}+a-9a-9+25.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{a-1-\frac{10+1}{a+1}}$$

Add 10 and 1 to get 11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{a-1-\frac{11}{a+1}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a-1 times \frac{a+1}{a+1}.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{\left(a-1\right)\left(a+1\right)}{a+1}-\frac{11}{a+1}}$$

Since \frac{\left(a-1\right)\left(a+1\right)}{a+1} and \frac{11}{a+1} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{\left(a-1\right)\left(a+1\right)-11}{a+1}}$$

Do the multiplications in \left(a-1\right)\left(a+1\right)-11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{a^{2}+a-a-1-11}{a+1}}$$

Combine like terms in a^{2}+a-a-1-11.

$$\frac{\frac{a^{2}-8a+16}{a+1}}{\frac{a^{2}-12}{a+1}}$$

Divide \frac{a^{2}-8a+16}{a+1} by \frac{a^{2}-12}{a+1} by multiplying \frac{a^{2}-8a+16}{a+1} by the reciprocal of \frac{a^{2}-12}{a+1}.

$$\frac{\left(a^{2}-8a+16\right)\left(a+1\right)}{\left(a+1\right)\left(a^{2}-12\right)}$$

Cancel out a+1 in both numerator and denominator.

$$\frac{a^{2}-8a+16}{a^{2}-12}$$