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Simplifying 14z2 + -3 = 21z + -7 Reorder the terms: -3 + 14z2 = 21z + -7 Reorder the terms: -3 + 14z2 = -7 + 21z Solving -3 + 14z2 = -7 + 21z Solving for variable 'z'. Reorder the terms: -3 + 7 + -21z + 14z2 = -7 + 21z + 7 + -21z Combine like terms: -3 + 7 = 4 4 + -21z + 14z2 = -7 + 21z + 7 + -21z Reorder the terms: 4 + -21z + 14z2 = -7 + 7 + 21z + -21z Combine like terms: -7 + 7 = 0 4 + -21z + 14z2 = 0 + 21z + -21z 4 + -21z + 14z2 = 21z + -21z Combine like terms: 21z + -21z = 0 4 + -21z + 14z2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 0.2857142857 + -1.5z + z2 = 0 Move the constant term to the right: Add '-0.2857142857' to each side of the equation. 0.2857142857 + -1.5z + -0.2857142857 + z2 = 0 + -0.2857142857 Reorder the terms: 0.2857142857 + -0.2857142857 + -1.5z + z2 = 0 + -0.2857142857 Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000 0.0000000000 + -1.5z + z2 = 0 + -0.2857142857 -1.5z + z2 = 0 + -0.2857142857 Combine like terms: 0 + -0.2857142857 = -0.2857142857 -1.5z + z2 = -0.2857142857 The z term is -1.5z. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5z + 0.5625 + z2 = -0.2857142857 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = -0.2857142857 + 0.5625 Combine like terms: -0.2857142857 + 0.5625 = 0.2767857143 0.5625 + -1.5z + z2 = 0.2767857143 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 0.2767857143 Calculate the square root of the right side: 0.526104281 Break this problem into two subproblems by setting (z + -0.75) equal to 0.526104281 and -0.526104281.Subproblem 1
z + -0.75 = 0.526104281 Simplifying z + -0.75 = 0.526104281 Reorder the terms: -0.75 + z = 0.526104281 Solving -0.75 + z = 0.526104281 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = 0.526104281 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 0.526104281 + 0.75 z = 0.526104281 + 0.75 Combine like terms: 0.526104281 + 0.75 = 1.276104281 z = 1.276104281 Simplifying z = 1.276104281Subproblem 2
z + -0.75 = -0.526104281 Simplifying z + -0.75 = -0.526104281 Reorder the terms: -0.75 + z = -0.526104281 Solving -0.75 + z = -0.526104281 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + z = -0.526104281 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -0.526104281 + 0.75 z = -0.526104281 + 0.75 Combine like terms: -0.526104281 + 0.75 = 0.223895719 z = 0.223895719 Simplifying z = 0.223895719Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.276104281, 0.223895719}
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