16Z^4-20Z^2+5=0

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Solution for 16Z^4-20Z^2+5=0 equation:


Simplifying
16Z4 + -20Z2 + 5 = 0

Reorder the terms:
5 + -20Z2 + 16Z4 = 0

Solving
5 + -20Z2 + 16Z4 = 0

Solving for variable 'Z'.

Begin completing the square.  Divide all terms by
16 the coefficient of the squared term: 

Divide each side by '16'.
0.3125 + -1.25Z2 + Z4 = 0

Move the constant term to the right:

Add '-0.3125' to each side of the equation.
0.3125 + -1.25Z2 + -0.3125 + Z4 = 0 + -0.3125

Reorder the terms:
0.3125 + -0.3125 + -1.25Z2 + Z4 = 0 + -0.3125

Combine like terms: 0.3125 + -0.3125 = 0.0000
0.0000 + -1.25Z2 + Z4 = 0 + -0.3125
-1.25Z2 + Z4 = 0 + -0.3125

Combine like terms: 0 + -0.3125 = -0.3125
-1.25Z2 + Z4 = -0.3125

The Z term is -1.25Z2.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25Z2 + 0.390625 + Z4 = -0.3125 + 0.390625

Reorder the terms:
0.390625 + -1.25Z2 + Z4 = -0.3125 + 0.390625

Combine like terms: -0.3125 + 0.390625 = 0.078125
0.390625 + -1.25Z2 + Z4 = 0.078125

Factor a perfect square on the left side:
(Z2 + -0.625)(Z2 + -0.625) = 0.078125

Calculate the square root of the right side: 0.279508497

Break this problem into two subproblems by setting 
(Z2 + -0.625) equal to 0.279508497 and -0.279508497.

Subproblem 1

Z2 + -0.625 = 0.279508497 Simplifying Z2 + -0.625 = 0.279508497 Reorder the terms: -0.625 + Z2 = 0.279508497 Solving -0.625 + Z2 = 0.279508497 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + Z2 = 0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + Z2 = 0.279508497 + 0.625 Z2 = 0.279508497 + 0.625 Combine like terms: 0.279508497 + 0.625 = 0.904508497 Z2 = 0.904508497 Simplifying Z2 = 0.904508497 Take the square root of each side: Z = {-0.951056516, 0.951056516}

Subproblem 2

Z2 + -0.625 = -0.279508497 Simplifying Z2 + -0.625 = -0.279508497 Reorder the terms: -0.625 + Z2 = -0.279508497 Solving -0.625 + Z2 = -0.279508497 Solving for variable 'Z'. Move all terms containing Z to the left, all other terms to the right. Add '0.625' to each side of the equation. -0.625 + 0.625 + Z2 = -0.279508497 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + Z2 = -0.279508497 + 0.625 Z2 = -0.279508497 + 0.625 Combine like terms: -0.279508497 + 0.625 = 0.345491503 Z2 = 0.345491503 Simplifying Z2 = 0.345491503 Take the square root of each side: Z = {-0.587785252, 0.587785252}

Solution

The solution to the problem is based on the solutions from the subproblems. Z = {-0.951056516, 0.951056516, -0.587785252, 0.587785252}

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