Solve 8/x^2-4x+9+5x/x-3

$$\frac { 8 } { x ^ { 2 } - 4 x + 9 } + \frac { 5 x } { x - 3 }$$

Evaluate

$$\frac{5x^{3}-20x^{2}+53x-24}{\left(x-3\right)\left(x^{2}-4x+9\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x^{2}-4x+9 and x-3 is \left(x-3\right)\left(x^{2}-4x+9\right). Multiply \frac{8}{x^{2}-4x+9} times \frac{x-3}{x-3}. Multiply \frac{5x}{x-3} times \frac{x^{2}-4x+9}{x^{2}-4x+9}.

$$\frac{8\left(x-3\right)}{\left(x-3\right)\left(x^{2}-4x+9\right)}+\frac{5x\left(x^{2}-4x+9\right)}{\left(x-3\right)\left(x^{2}-4x+9\right)}$$

Since \frac{8\left(x-3\right)}{\left(x-3\right)\left(x^{2}-4x+9\right)} and \frac{5x\left(x^{2}-4x+9\right)}{\left(x-3\right)\left(x^{2}-4x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{8\left(x-3\right)+5x\left(x^{2}-4x+9\right)}{\left(x-3\right)\left(x^{2}-4x+9\right)}$$

Do the multiplications in 8\left(x-3\right)+5x\left(x^{2}-4x+9\right).

$$\frac{8x-24+5x^{3}-20x^{2}+45x}{\left(x-3\right)\left(x^{2}-4x+9\right)}$$

Combine like terms in 8x-24+5x^{3}-20x^{2}+45x.

$$\frac{53x-24+5x^{3}-20x^{2}}{\left(x-3\right)\left(x^{2}-4x+9\right)}$$

Expand \left(x-3\right)\left(x^{2}-4x+9\right).

$$\frac{53x-24+5x^{3}-20x^{2}}{x^{3}-7x^{2}+21x-27}$$

Differentiate w.r.t. x

$$\frac{-15x^{4}+104x^{3}-382x^{2}+744x-927}{\left(x-3\right)^{2}\left(x^{4}-8x^{3}+34x^{2}-72x+81\right)}$$