Solve frac{x/frac{3{2}}}{x/sqrt{5/2-x}}

$$\frac{ \frac{ x }{ \frac{ 3 }{ 2 } } }{ \frac{ x }{ \frac{ \sqrt{ 5 } }{ 2 } -x } }$$

Evaluate

$$\frac{-2x+\sqrt{5}}{3}$$

Short Solution Steps

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{2}{2}.

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

Since \frac{\sqrt{5}}{2} and \frac{2x}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times \frac{\sqrt{5}-2x}{2}}{\frac{3}{2}x}$$

Express x\times \frac{\sqrt{5}-2x}{2} as a single fraction.

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

Express \frac{\frac{x\left(\sqrt{5}-2x\right)}{2}}{\frac{3}{2}x} as a single fraction.

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

Cancel out x in both numerator and denominator.

$$\frac{-2x+\sqrt{5}}{\frac{3}{2}\times 2}$$

Cancel out 2 and 2.

$$\frac{-2x+\sqrt{5}}{3}$$

Differentiate w.r.t. x

$$-\frac{2}{3} = -0.6666666666666666$$