Solve a^2-5/a-sqrt{5}

$$\frac { a ^ { 2 } - 5 } { a - \sqrt { 5 } }$$

Evaluate

$$\frac{a^{2}-5}{a-\sqrt{5}}$$

Solution Steps

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

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

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

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

The square of \sqrt{5} is 5.

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

Cancel out a^{2}-5 in both numerator and denominator.

$$a+\sqrt{5}$$

Differentiate w.r.t. a

$$1$$