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Simplifying -2z2 + -2z + 5 = 0 Reorder the terms: 5 + -2z + -2z2 = 0 Solving 5 + -2z + -2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + z + z2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + z + 2.5 + z2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + z + z2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z + z2 = 0 + 2.5 z + z2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 z + z2 = 2.5 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 = 2.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 2.5 + 0.25 Combine like terms: 2.5 + 0.25 = 2.75 1.25 + z2 = 2.75 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (z + 0.5) equal to 1.658312395 and -1.658312395.Subproblem 1
z + 0.5 = 1.658312395 Simplifying z + 0.5 = 1.658312395 Reorder the terms: 0.5 + z = 1.658312395 Solving 0.5 + z = 1.658312395 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 = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.658312395 + -0.5 z = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 z = 1.158312395 Simplifying z = 1.158312395Subproblem 2
z + 0.5 = -1.658312395 Simplifying z + 0.5 = -1.658312395 Reorder the terms: 0.5 + z = -1.658312395 Solving 0.5 + z = -1.658312395 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 = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.658312395 + -0.5 z = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 z = -2.158312395 Simplifying z = -2.158312395Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.158312395, -2.158312395}
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