z^4-8z^2+11=0

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Solution for z^4-8z^2+11=0 equation:


Simplifying
z4 + -8z2 + 11 = 0

Reorder the terms:
11 + -8z2 + z4 = 0

Solving
11 + -8z2 + z4 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-11' to each side of the equation.
11 + -8z2 + -11 + z4 = 0 + -11

Reorder the terms:
11 + -11 + -8z2 + z4 = 0 + -11

Combine like terms: 11 + -11 = 0
0 + -8z2 + z4 = 0 + -11
-8z2 + z4 = 0 + -11

Combine like terms: 0 + -11 = -11
-8z2 + z4 = -11

The z term is -8z2.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z2 + 16 + z4 = -11 + 16

Reorder the terms:
16 + -8z2 + z4 = -11 + 16

Combine like terms: -11 + 16 = 5
16 + -8z2 + z4 = 5

Factor a perfect square on the left side:
(z2 + -4)(z2 + -4) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(z2 + -4) equal to 2.236067978 and -2.236067978.

Subproblem 1

z2 + -4 = 2.236067978 Simplifying z2 + -4 = 2.236067978 Reorder the terms: -4 + z2 = 2.236067978 Solving -4 + z2 = 2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z2 = 2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z2 = 2.236067978 + 4 z2 = 2.236067978 + 4 Combine like terms: 2.236067978 + 4 = 6.236067978 z2 = 6.236067978 Simplifying z2 = 6.236067978 Take the square root of each side: z = {-2.497212041, 2.497212041}

Subproblem 2

z2 + -4 = -2.236067978 Simplifying z2 + -4 = -2.236067978 Reorder the terms: -4 + z2 = -2.236067978 Solving -4 + z2 = -2.236067978 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4' to each side of the equation. -4 + 4 + z2 = -2.236067978 + 4 Combine like terms: -4 + 4 = 0 0 + z2 = -2.236067978 + 4 z2 = -2.236067978 + 4 Combine like terms: -2.236067978 + 4 = 1.763932022 z2 = 1.763932022 Simplifying z2 = 1.763932022 Take the square root of each side: z = {-1.328131026, 1.328131026}

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-2.497212041, 2.497212041, -1.328131026, 1.328131026}

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