Solve 5/x^2-5x+6-4/x^2-6x+9

$$\frac { 5 } { x ^ { 2 } - 5 x + 6 } - \frac { 4 } { x ^ { 2 } - 6 x + 9 }$$

Evaluate

$$\frac{x-7}{\left(x-2\right)\left(x-3\right)^{2}}$$

Short Solution Steps

Factor x^{2}-5x+6. Factor x^{2}-6x+9.

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

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

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

Since \frac{5\left(x-3\right)}{\left(x-2\right)\left(x-3\right)^{2}} and \frac{4\left(x-2\right)}{\left(x-2\right)\left(x-3\right)^{2}} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 5\left(x-3\right)-4\left(x-2\right).

$$\frac{5x-15-4x+8}{\left(x-2\right)\left(x-3\right)^{2}}$$

Combine like terms in 5x-15-4x+8.

$$\frac{x-7}{\left(x-2\right)\left(x-3\right)^{2}}$$

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

$$\frac{x-7}{x^{3}-8x^{2}+21x-18}$$

Differentiate w.r.t. x

$$-\frac{\left(4x-23\right)^{2}-185}{8\left(x-2\right)^{2}\left(x-3\right)^{3}}$$