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Simplifying -2z2 + -2z + 8 = 0 Reorder the terms: 8 + -2z + -2z2 = 0 Solving 8 + -2z + -2z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(4 + -1z + -1z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + -1z + -1z2)' equal to zero and attempt to solve: Simplifying 4 + -1z + -1z2 = 0 Solving 4 + -1z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -4 + z + z2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + z + 4 + z2 = 0 + 4 Reorder the terms: -4 + 4 + z + z2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + z + z2 = 0 + 4 z + z2 = 0 + 4 Combine like terms: 0 + 4 = 4 z + z2 = 4 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 = 4 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + z2 = 4 + 0.25 Combine like terms: 4 + 0.25 = 4.25 1.25 + z2 = 4.25 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (z + 0.5) equal to 2.061552813 and -2.061552813.Subproblem 1
z + 0.5 = 2.061552813 Simplifying z + 0.5 = 2.061552813 Reorder the terms: 0.5 + z = 2.061552813 Solving 0.5 + z = 2.061552813 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 = 2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 2.061552813 + -0.5 z = 2.061552813 + -0.5 Combine like terms: 2.061552813 + -0.5 = 1.561552813 z = 1.561552813 Simplifying z = 1.561552813Subproblem 2
z + 0.5 = -2.061552813 Simplifying z + 0.5 = -2.061552813 Reorder the terms: 0.5 + z = -2.061552813 Solving 0.5 + z = -2.061552813 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 = -2.061552813 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -2.061552813 + -0.5 z = -2.061552813 + -0.5 Combine like terms: -2.061552813 + -0.5 = -2.561552813 z = -2.561552813 Simplifying z = -2.561552813Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.561552813, -2.561552813}Solution
z = {1.561552813, -2.561552813}
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