4z^2+4z-2=0

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


Simplifying
4z2 + 4z + -2 = 0

Reorder the terms:
-2 + 4z + 4z2 = 0

Solving
-2 + 4z + 4z2 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-1 + 2z + 2z2) = 0

Ignore the factor 2.

Subproblem 1

Set the factor '(-1 + 2z + 2z2)' equal to zero and attempt to solve: Simplifying -1 + 2z + 2z2 = 0 Solving -1 + 2z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + z + z2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + z + 0.5 + z2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + z + z2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z + z2 = 0 + 0.5 z + z2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 z + z2 = 0.5 The z term is z. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + z2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + z2 = 0.75 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (z + 0.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

z + 0.5 = 0.866025404 Simplifying z + 0.5 = 0.866025404 Reorder the terms: 0.5 + z = 0.866025404 Solving 0.5 + z = 0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 0.866025404 + -0.5 z = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 z = 0.366025404 Simplifying z = 0.366025404

Subproblem 2

z + 0.5 = -0.866025404 Simplifying z + 0.5 = -0.866025404 Reorder the terms: 0.5 + z = -0.866025404 Solving 0.5 + z = -0.866025404 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -0.866025404 + -0.5 z = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 z = -1.366025404 Simplifying z = -1.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.366025404, -1.366025404}

Solution

z = {0.366025404, -1.366025404}

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