Solve w/w^2-24w+2+6

$$\frac { w } { w ^ { 2 } - 24 w + 2 } + 6$$

Evaluate

$$\frac{6w^{2}-143w+12}{w^{2}-24w+2}$$

Short Solution Steps

Factor w^{2}-24w+2.

$$\frac{w}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}+6$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 6 times \frac{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}.

$$\frac{w}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}+\frac{6\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}$$

Since \frac{w}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)} and \frac{6\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)} have the same denominator, add them by adding their numerators.

$$\frac{w+6\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}$$

Do the multiplications in w+6\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right).

$$\frac{w+6w^{2}+6w\sqrt{142}-72w-6\sqrt{142}w+12-72w}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}$$

Combine like terms in w+6w^{2}+6w\sqrt{142}-72w-6\sqrt{142}w+12-72w.

$$\frac{-143w+6w^{2}+12}{\left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right)}$$

Expand \left(w-\left(\sqrt{142}+12\right)\right)\left(w-\left(-\sqrt{142}+12\right)\right).

$$\frac{-143w+6w^{2}+12}{w^{2}-24w-\left(\sqrt{142}\right)^{2}+144}$$

The square of \sqrt{142} is 142.

$$\frac{-143w+6w^{2}+12}{w^{2}-24w-142+144}$$

Add -142 and 144 to get 2.

$$\frac{-143w+6w^{2}+12}{w^{2}-24w+2}$$

Differentiate w.r.t. w

$$\frac{2-w^{2}}{w^{4}-48w^{3}+580w^{2}-96w+4}$$