Solve 5/x^2-6x+8-2/x^2-8x+16

$$\frac { 5 } { x ^ { 2 } - 6 x + 8 } - \frac { 2 } { x ^ { 2 } - 8 x + 16 }$$

Evaluate

$$\frac{3x-16}{\left(x-2\right)\left(x-4\right)^{2}}$$

Short Solution Steps

Factor x^{2}-6x+8. Factor x^{2}-8x+16.

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

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

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

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

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

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

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

Combine like terms in 5x-20-2x+4.

$$\frac{3x-16}{\left(x-2\right)\left(x-4\right)^{2}}$$

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

$$\frac{3x-16}{x^{3}-10x^{2}+32x-32}$$

Differentiate w.r.t. x

$$-\frac{\left(6x-27\right)^{2}-105}{6\left(x-2\right)^{2}\left(x-4\right)^{3}}$$