Solve (a+sqrt{5})(a-sqrt{5})-(sqrt{3}-a)^2

$$( a + \sqrt { 5 } ) ( a - \sqrt { 5 } ) - ( \sqrt { 3 } - a ) ^ { 2 }$$

Evaluate

$$2\left(\sqrt{3}a-4\right)$$

Solution Steps

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}.

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

The square of \sqrt{5} is 5.

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

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(\sqrt{3}-a\right)^{2}.

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

The square of \sqrt{3} is 3.

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

To find the opposite of 3-2\sqrt{3}a+a^{2}, find the opposite of each term.

$$a^{2}-5-3+2\sqrt{3}a-a^{2}$$

Subtract 3 from -5 to get -8.

$$a^{2}-8+2\sqrt{3}a-a^{2}$$

Combine a^{2} and -a^{2} to get 0.

$$-8+2\sqrt{3}a$$

Differentiate w.r.t. a

$$2 \sqrt{3} = 3.464101615$$