Solve the fourth power equation z^4-12z^2+11=0

This equation is also called biquadratic
equation.

To solve this equation, we'll reduce it to a
quadratic equation by replacing z^2 = x.

We'll re-write the
equation in x:

x^2 - 12x + 11 =
0

We'll apply quadratic
formula:

x1 = [12+
sqrt(144-44)]/2

x1 =
(12+10)/2

x1 = 11

x2 =
(12-10)/2

x2 = 1

But, we'll
have to find z1,z2,z3,z4.

z^2 =
x1

z^2 = 11

z1 = sqrt 11 and
z2 = -sqrt 11

z^2 = x2

z^2 =
1

z3 = -1 and z4 =
1

The solutions of the biquadratic equation
are: {-sqrt11 ; -1 ; 1 ; sqrt11}.