Solve the equation with square root coefficient

a^3+2\sqrt{5}a^2+5a+\sqrt{5}-1 =0

Solve the equation with square root coefficient

a^3+2\sqrt{11}a^2+11a+\sqrt{11}+1 =0

Solve the equation with square root coefficient

a^3+2\sqrt{5}a^2+5a+\sqrt{5}+1 =0

Solve the equation with square root coefficient

a^3+2\sqrt{7}a^2+7a+\sqrt{7}+1 =0

If a is a root for the following equation

x^2+\sqrt{3}x-2 = 0

Find the value of

\dfrac{a^4}{a^8+a^6+2a^4+4a^2+16}

Let n be a rational number such that the equation

x^2-(\sqrt{3}+1)x+\sqrt{3}n-6 = 0

has an integer root. Find the value of n.

One of the roots of a quadratic equation x^2-(\tan \alpha+\cot \alpha )x+1 = 0 in terms of x is 2+\sqrt{3} . Evaluate \sin \alpha \cdotp \cos \alpha ,