z^4-10z^2+1=0

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Solution for z^4-10z^2+1=0 equation:


Simplifying
z4 + -10z2 + 1 = 0

Reorder the terms:
1 + -10z2 + z4 = 0

Solving
1 + -10z2 + z4 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -10z2 + -1 + z4 = 0 + -1

Reorder the terms:
1 + -1 + -10z2 + z4 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -10z2 + z4 = 0 + -1
-10z2 + z4 = 0 + -1

Combine like terms: 0 + -1 = -1
-10z2 + z4 = -1

The z term is -10z2.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10z2 + 25 + z4 = -1 + 25

Reorder the terms:
25 + -10z2 + z4 = -1 + 25

Combine like terms: -1 + 25 = 24
25 + -10z2 + z4 = 24

Factor a perfect square on the left side:
(z2 + -5)(z2 + -5) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(z2 + -5) equal to 4.898979486 and -4.898979486.

Subproblem 1

z2 + -5 = 4.898979486 Simplifying z2 + -5 = 4.898979486 Reorder the terms: -5 + z2 = 4.898979486 Solving -5 + z2 = 4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z2 = 4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = 4.898979486 + 5 z2 = 4.898979486 + 5 Combine like terms: 4.898979486 + 5 = 9.898979486 z2 = 9.898979486 Simplifying z2 = 9.898979486 Take the square root of each side: z = {-3.14626437, 3.14626437}

Subproblem 2

z2 + -5 = -4.898979486 Simplifying z2 + -5 = -4.898979486 Reorder the terms: -5 + z2 = -4.898979486 Solving -5 + z2 = -4.898979486 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z2 = -4.898979486 + 5 Combine like terms: -5 + 5 = 0 0 + z2 = -4.898979486 + 5 z2 = -4.898979486 + 5 Combine like terms: -4.898979486 + 5 = 0.101020514 z2 = 0.101020514 Simplifying z2 = 0.101020514 Take the square root of each side: z = {-0.317837245, 0.317837245}

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-3.14626437, 3.14626437, -0.317837245, 0.317837245}

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