Solve frac{sqrt{10}}{2x^2-sqrt{2}x}:frac{3sqrt{5}}{2x^3-x}

$$\frac { \sqrt { 10 } } { 2 x ^ { 2 } - \sqrt { 2 } x } : \frac { 3 \sqrt { 5 } } { 2 x ^ { 3 } - x }$$

Evaluate

$$\frac{\sqrt{2}x+1}{3}$$

Short Solution Steps

Divide \frac{\sqrt{10}}{2x^{2}-\sqrt{2}x} by \frac{3\sqrt{5}}{2x^{3}-x} by multiplying \frac{\sqrt{10}}{2x^{2}-\sqrt{2}x} by the reciprocal of \frac{3\sqrt{5}}{2x^{3}-x}.

$$\frac{\sqrt{10}\left(2x^{3}-x\right)}{\left(2x^{2}-\sqrt{2}x\right)\times 3\sqrt{5}}$$

Factor the expressions that are not already factored.

$$\frac{\sqrt{10}x\left(2x^{2}-1\right)}{3\sqrt{2}\sqrt{5}x\left(\sqrt{2}x-1\right)}$$

Cancel out x in both numerator and denominator.

$$\frac{\sqrt{10}\left(2x^{2}-1\right)}{3\sqrt{2}\sqrt{5}\left(\sqrt{2}x-1\right)}$$

Expand the expression.

$$\frac{2\sqrt{10}x^{2}-\sqrt{10}}{6\sqrt{5}x-3\sqrt{10}}$$

Rationalize the denominator of \frac{2\sqrt{10}x^{2}-\sqrt{10}}{6\sqrt{5}x-3\sqrt{10}} by multiplying numerator and denominator by 6\sqrt{5}x+3\sqrt{10}.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{\left(6\sqrt{5}x-3\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}$$

Consider \left(6\sqrt{5}x-3\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{\left(6\sqrt{5}x\right)^{2}-\left(-3\sqrt{10}\right)^{2}}$$

Expand \left(6\sqrt{5}x\right)^{2}.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{6^{2}\left(\sqrt{5}\right)^{2}x^{2}-\left(-3\sqrt{10}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{36\left(\sqrt{5}\right)^{2}x^{2}-\left(-3\sqrt{10}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{36\times 5x^{2}-\left(-3\sqrt{10}\right)^{2}}$$

Multiply 36 and 5 to get 180.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{180x^{2}-\left(-3\sqrt{10}\right)^{2}}$$

Expand \left(-3\sqrt{10}\right)^{2}.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{180x^{2}-\left(-3\right)^{2}\left(\sqrt{10}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{180x^{2}-9\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{180x^{2}-9\times 10}$$

Multiply 9 and 10 to get 90.

$$\frac{\left(2\sqrt{10}x^{2}-\sqrt{10}\right)\left(6\sqrt{5}x+3\sqrt{10}\right)}{180x^{2}-90}$$

Factor the expressions that are not already factored.

$$\frac{3\sqrt{10}\left(2\sqrt{5}x+\sqrt{10}\right)\left(2x^{2}-1\right)}{90\left(2x^{2}-1\right)}$$

Cancel out 3\left(2x^{2}-1\right) in both numerator and denominator.

$$\frac{\sqrt{10}\left(2\sqrt{5}x+\sqrt{10}\right)}{30}$$

Expand the expression.

$$\frac{10\sqrt{2}x+10}{30}$$

Differentiate w.r.t. x

$$\frac{\sqrt{2}}{3} = 0.47140452079103173$$