Solve frac{5x-sqrt{3}x^2}{sqrt{3}+x}

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

Evaluate

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

Short Solution Steps

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

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

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

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

The square of \sqrt{3} is 3.

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

Use the distributive property to multiply 5x-\sqrt{3}x^{2} by \sqrt{3}-x.

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

Use the distributive property to multiply 5x-\sqrt{3}x^{2} by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Multiply -1 and 3 to get -3.

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

Use the distributive property to multiply -1 by 5x-\sqrt{3}x^{2}.

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

Use the distributive property to multiply -5x+\sqrt{3}x^{2} by x.

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

Combine -3x^{2} and -5x^{2} to get -8x^{2}.

$$\frac{5x\sqrt{3}-8x^{2}+\sqrt{3}x^{3}}{3-x^{2}}$$

Factor

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