Solve x^2-10/x-sqrt{12}

$$\frac { x ^ { 2 } - 10 } { x - \sqrt { 12 } }$$

Evaluate

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

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

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

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

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

Consider \left(x-2\sqrt{3}\right)\left(x+2\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}.

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

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

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

Calculate -2 to the power of 2 and get 4.

$$\frac{\left(x^{2}-10\right)\left(x+2\sqrt{3}\right)}{x^{2}-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(x^{2}-10\right)\left(x+2\sqrt{3}\right)}{x^{2}-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(x^{2}-10\right)\left(x+2\sqrt{3}\right)}{x^{2}-12}$$

Use the distributive property to multiply x^{2}-10 by x+2\sqrt{3}.

$$\frac{x^{3}+2x^{2}\sqrt{3}-10x-20\sqrt{3}}{x^{2}-12}$$

Differentiate w.r.t. x

$$\frac{x^{2}-4\sqrt{3}x+10}{\left(x-2\sqrt{3}\right)^{2}}$$