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Simplifying -7z2 + 7z + 2 = 0 Reorder the terms: 2 + 7z + -7z2 = 0 Solving 2 + 7z + -7z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -0.2857142857 + -1z + z2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -1z + 0.2857142857 + z2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -1z + z2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -1z + z2 = 0 + 0.2857142857 -1z + z2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -1z + z2 = 0.2857142857 The z term is -1z. 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. -1z + 0.25 + z2 = 0.2857142857 + 0.25 Reorder the terms: 0.25 + -1z + z2 = 0.2857142857 + 0.25 Combine like terms: 0.2857142857 + 0.25 = 0.5357142857 0.25 + -1z + z2 = 0.5357142857 Factor a perfect square on the left side: (z + -0.5)(z + -0.5) = 0.5357142857 Calculate the square root of the right side: 0.731925055 Break this problem into two subproblems by setting (z + -0.5) equal to 0.731925055 and -0.731925055.Subproblem 1
z + -0.5 = 0.731925055 Simplifying z + -0.5 = 0.731925055 Reorder the terms: -0.5 + z = 0.731925055 Solving -0.5 + z = 0.731925055 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 = 0.731925055 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = 0.731925055 + 0.5 z = 0.731925055 + 0.5 Combine like terms: 0.731925055 + 0.5 = 1.231925055 z = 1.231925055 Simplifying z = 1.231925055Subproblem 2
z + -0.5 = -0.731925055 Simplifying z + -0.5 = -0.731925055 Reorder the terms: -0.5 + z = -0.731925055 Solving -0.5 + z = -0.731925055 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 = -0.731925055 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + z = -0.731925055 + 0.5 z = -0.731925055 + 0.5 Combine like terms: -0.731925055 + 0.5 = -0.231925055 z = -0.231925055 Simplifying z = -0.231925055Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.231925055, -0.231925055}
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