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Simplifying 7z + 8 = 8z2 Reorder the terms: 8 + 7z = 8z2 Solving 8 + 7z = 8z2 Solving for variable 'z'. Combine like terms: 8z2 + -8z2 = 0 8 + 7z + -8z2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1 + -0.875z + z2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.875z + 1 + z2 = 0 + 1 Reorder the terms: -1 + 1 + -0.875z + z2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.875z + z2 = 0 + 1 -0.875z + z2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.875z + z2 = 1 The z term is -0.875z. Take half its coefficient (-0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. -0.875z + 0.19140625 + z2 = 1 + 0.19140625 Reorder the terms: 0.19140625 + -0.875z + z2 = 1 + 0.19140625 Combine like terms: 1 + 0.19140625 = 1.19140625 0.19140625 + -0.875z + z2 = 1.19140625 Factor a perfect square on the left side: (z + -0.4375)(z + -0.4375) = 1.19140625 Calculate the square root of the right side: 1.091515575 Break this problem into two subproblems by setting (z + -0.4375) equal to 1.091515575 and -1.091515575.Subproblem 1
z + -0.4375 = 1.091515575 Simplifying z + -0.4375 = 1.091515575 Reorder the terms: -0.4375 + z = 1.091515575 Solving -0.4375 + z = 1.091515575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + z = 1.091515575 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + z = 1.091515575 + 0.4375 z = 1.091515575 + 0.4375 Combine like terms: 1.091515575 + 0.4375 = 1.529015575 z = 1.529015575 Simplifying z = 1.529015575Subproblem 2
z + -0.4375 = -1.091515575 Simplifying z + -0.4375 = -1.091515575 Reorder the terms: -0.4375 + z = -1.091515575 Solving -0.4375 + z = -1.091515575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.4375' to each side of the equation. -0.4375 + 0.4375 + z = -1.091515575 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + z = -1.091515575 + 0.4375 z = -1.091515575 + 0.4375 Combine like terms: -1.091515575 + 0.4375 = -0.654015575 z = -0.654015575 Simplifying z = -0.654015575Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.529015575, -0.654015575}
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