Solve 7*2/x+sqrt{3}

$$7 \times \frac{ 2 }{ x+ \sqrt{ 3 } }$$

Evaluate

$$\frac{14}{x+\sqrt{3}}$$

Short Solution Steps

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

$$7\times \frac{2\left(x-\sqrt{3}\right)}{\left(x+\sqrt{3}\right)\left(x-\sqrt{3}\right)}$$

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

$$7\times \frac{2\left(x-\sqrt{3}\right)}{x^{2}-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$7\times \frac{2\left(x-\sqrt{3}\right)}{x^{2}-3}$$

Express 7\times \frac{2\left(x-\sqrt{3}\right)}{x^{2}-3} as a single fraction.

$$\frac{7\times 2\left(x-\sqrt{3}\right)}{x^{2}-3}$$

Multiply 7 and 2 to get 14.

$$\frac{14\left(x-\sqrt{3}\right)}{x^{2}-3}$$

Use the distributive property to multiply 14 by x-\sqrt{3}.

$$\frac{14x-14\sqrt{3}}{x^{2}-3}$$

Differentiate w.r.t. x

$$-\frac{14}{\left(x+\sqrt{3}\right)^{2}}$$