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Simplifying 14z + 8z2 = 27 Solving 14z + 8z2 = 27 Solving for variable 'z'. Reorder the terms: -27 + 14z + 8z2 = 27 + -27 Combine like terms: 27 + -27 = 0 -27 + 14z + 8z2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.375 + 1.75z + z2 = 0 Move the constant term to the right: Add '3.375' to each side of the equation. -3.375 + 1.75z + 3.375 + z2 = 0 + 3.375 Reorder the terms: -3.375 + 3.375 + 1.75z + z2 = 0 + 3.375 Combine like terms: -3.375 + 3.375 = 0.000 0.000 + 1.75z + z2 = 0 + 3.375 1.75z + z2 = 0 + 3.375 Combine like terms: 0 + 3.375 = 3.375 1.75z + z2 = 3.375 The z term is 1.75z. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75z + 0.765625 + z2 = 3.375 + 0.765625 Reorder the terms: 0.765625 + 1.75z + z2 = 3.375 + 0.765625 Combine like terms: 3.375 + 0.765625 = 4.140625 0.765625 + 1.75z + z2 = 4.140625 Factor a perfect square on the left side: (z + 0.875)(z + 0.875) = 4.140625 Calculate the square root of the right side: 2.034852575 Break this problem into two subproblems by setting (z + 0.875) equal to 2.034852575 and -2.034852575.Subproblem 1
z + 0.875 = 2.034852575 Simplifying z + 0.875 = 2.034852575 Reorder the terms: 0.875 + z = 2.034852575 Solving 0.875 + z = 2.034852575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + z = 2.034852575 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + z = 2.034852575 + -0.875 z = 2.034852575 + -0.875 Combine like terms: 2.034852575 + -0.875 = 1.159852575 z = 1.159852575 Simplifying z = 1.159852575Subproblem 2
z + 0.875 = -2.034852575 Simplifying z + 0.875 = -2.034852575 Reorder the terms: 0.875 + z = -2.034852575 Solving 0.875 + z = -2.034852575 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + z = -2.034852575 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + z = -2.034852575 + -0.875 z = -2.034852575 + -0.875 Combine like terms: -2.034852575 + -0.875 = -2.909852575 z = -2.909852575 Simplifying z = -2.909852575Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.159852575, -2.909852575}
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