z^4-45z^2=-4

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Solution for z^4-45z^2=-4 equation:


Simplifying
z4 + -45z2 = -4

Reorder the terms:
-45z2 + z4 = -4

Solving
-45z2 + z4 = -4

Solving for variable 'z'.

Reorder the terms:
4 + -45z2 + z4 = -4 + 4

Combine like terms: -4 + 4 = 0
4 + -45z2 + z4 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + -45z2 + -4 + z4 = 0 + -4

Reorder the terms:
4 + -4 + -45z2 + z4 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + -45z2 + z4 = 0 + -4
-45z2 + z4 = 0 + -4

Combine like terms: 0 + -4 = -4
-45z2 + z4 = -4

The z term is -45z2.  Take half its coefficient (-22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
-45z2 + 506.25 + z4 = -4 + 506.25

Reorder the terms:
506.25 + -45z2 + z4 = -4 + 506.25

Combine like terms: -4 + 506.25 = 502.25
506.25 + -45z2 + z4 = 502.25

Factor a perfect square on the left side:
(z2 + -22.5)(z2 + -22.5) = 502.25

Calculate the square root of the right side: 22.410934831

Break this problem into two subproblems by setting 
(z2 + -22.5) equal to 22.410934831 and -22.410934831.

Subproblem 1

z2 + -22.5 = 22.410934831 Simplifying z2 + -22.5 = 22.410934831 Reorder the terms: -22.5 + z2 = 22.410934831 Solving -22.5 + z2 = 22.410934831 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + z2 = 22.410934831 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + z2 = 22.410934831 + 22.5 z2 = 22.410934831 + 22.5 Combine like terms: 22.410934831 + 22.5 = 44.910934831 z2 = 44.910934831 Simplifying z2 = 44.910934831 Take the square root of each side: z = {-6.701562119, 6.701562119}

Subproblem 2

z2 + -22.5 = -22.410934831 Simplifying z2 + -22.5 = -22.410934831 Reorder the terms: -22.5 + z2 = -22.410934831 Solving -22.5 + z2 = -22.410934831 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + z2 = -22.410934831 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + z2 = -22.410934831 + 22.5 z2 = -22.410934831 + 22.5 Combine like terms: -22.410934831 + 22.5 = 0.089065169 z2 = 0.089065169 Simplifying z2 = 0.089065169 Take the square root of each side: z = {-0.298437881, 0.298437881}

Solution

The solution to the problem is based on the solutions from the subproblems. z = {-6.701562119, 6.701562119, -0.298437881, 0.298437881}

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