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Simplifying 2z2 + 2z = 6 Reorder the terms: 2z + 2z2 = 6 Solving 2z + 2z2 = 6 Solving for variable 'z'. Reorder the terms: -6 + 2z + 2z2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 2z + 2z2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + z + z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + z + z2)' equal to zero and attempt to solve: Simplifying -3 + z + z2 = 0 Solving -3 + z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + z + 3 + z2 = 0 + 3 Reorder the terms: -3 + 3 + z + z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + z + z2 = 0 + 3 z + z2 = 0 + 3 Combine like terms: 0 + 3 = 3 z + z2 = 3 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. z + 0.25 + z2 = 3 + 0.25 Reorder the terms: 0.25 + z + z2 = 3 + 0.25 Combine like terms: 3 + 0.25 = 3.25 0.25 + z + z2 = 3.25 Factor a perfect square on the left side: (z + 0.5)(z + 0.5) = 3.25 Calculate the square root of the right side: 1.802775638 Break this problem into two subproblems by setting (z + 0.5) equal to 1.802775638 and -1.802775638.Subproblem 1
z + 0.5 = 1.802775638 Simplifying z + 0.5 = 1.802775638 Reorder the terms: 0.5 + z = 1.802775638 Solving 0.5 + z = 1.802775638 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 = 1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 1.802775638 + -0.5 z = 1.802775638 + -0.5 Combine like terms: 1.802775638 + -0.5 = 1.302775638 z = 1.302775638 Simplifying z = 1.302775638Subproblem 2
z + 0.5 = -1.802775638 Simplifying z + 0.5 = -1.802775638 Reorder the terms: 0.5 + z = -1.802775638 Solving 0.5 + z = -1.802775638 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 = -1.802775638 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -1.802775638 + -0.5 z = -1.802775638 + -0.5 Combine like terms: -1.802775638 + -0.5 = -2.302775638 z = -2.302775638 Simplifying z = -2.302775638Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.302775638, -2.302775638}Solution
z = {1.302775638, -2.302775638}
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