Solve 4y+9/y^2+2y-24+7/y^2+5y-6

$$\frac { 4 y + 9 } { y ^ { 2 } + 2 y - 24 } + \frac { 7 } { y ^ { 2 } + 5 y - 6 }$$

Evaluate

$$\frac{4y^{2}+12y-37}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Short Solution Steps

Factor y^{2}+2y-24. Factor y^{2}+5y-6.

$$\frac{4y+9}{\left(y-4\right)\left(y+6\right)}+\frac{7}{\left(y-1\right)\left(y+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-4\right)\left(y+6\right) and \left(y-1\right)\left(y+6\right) is \left(y-4\right)\left(y-1\right)\left(y+6\right). Multiply \frac{4y+9}{\left(y-4\right)\left(y+6\right)} times \frac{y-1}{y-1}. Multiply \frac{7}{\left(y-1\right)\left(y+6\right)} times \frac{y-4}{y-4}.

$$\frac{\left(4y+9\right)\left(y-1\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}+\frac{7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Since \frac{\left(4y+9\right)\left(y-1\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)} and \frac{7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4y+9\right)\left(y-1\right)+7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Do the multiplications in \left(4y+9\right)\left(y-1\right)+7\left(y-4\right).

$$\frac{4y^{2}-4y+9y-9+7y-28}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Combine like terms in 4y^{2}-4y+9y-9+7y-28.

$$\frac{4y^{2}+12y-37}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Expand \left(y-4\right)\left(y-1\right)\left(y+6\right).

$$\frac{4y^{2}+12y-37}{y^{3}+y^{2}-26y+24}$$

Expand

$$\frac{4y^{2}+12y-37}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Short Solution Steps

Factor y^{2}+2y-24. Factor y^{2}+5y-6.

$$\frac{4y+9}{\left(y-4\right)\left(y+6\right)}+\frac{7}{\left(y-1\right)\left(y+6\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(y-4\right)\left(y+6\right) and \left(y-1\right)\left(y+6\right) is \left(y-4\right)\left(y-1\right)\left(y+6\right). Multiply \frac{4y+9}{\left(y-4\right)\left(y+6\right)} times \frac{y-1}{y-1}. Multiply \frac{7}{\left(y-1\right)\left(y+6\right)} times \frac{y-4}{y-4}.

$$\frac{\left(4y+9\right)\left(y-1\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}+\frac{7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Since \frac{\left(4y+9\right)\left(y-1\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)} and \frac{7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4y+9\right)\left(y-1\right)+7\left(y-4\right)}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Do the multiplications in \left(4y+9\right)\left(y-1\right)+7\left(y-4\right).

$$\frac{4y^{2}-4y+9y-9+7y-28}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Combine like terms in 4y^{2}-4y+9y-9+7y-28.

$$\frac{4y^{2}+12y-37}{\left(y-4\right)\left(y-1\right)\left(y+6\right)}$$

Expand \left(y-4\right)\left(y-1\right)\left(y+6\right).

$$\frac{4y^{2}+12y-37}{y^{3}+y^{2}-26y+24}$$