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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 = -10 -10 + -21z + 14z2 = 7 + 21z + -7 + -21z Reorder the terms: -10 + -21z + 14z2 = 7 + -7 + 21z + -21z Combine like terms: 7 + -7 = 0 -10 + -21z + 14z2 = 0 + 21z + -21z -10 + -21z + 14z2 = 21z + -21z Combine like terms: 21z + -21z = 0 -10 + -21z + 14z2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.7142857143 + -1.5z + z2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + -1.5z + 0.7142857143 + z2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + -1.5z + z2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + -1.5z + z2 = 0 + 0.7142857143 -1.5z + z2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 -1.5z + z2 = 0.7142857143 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 = 0.7142857143 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 0.7142857143 + 0.5625 Combine like terms: 0.7142857143 + 0.5625 = 1.2767857143 0.5625 + -1.5z + z2 = 1.2767857143 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 1.2767857143 Calculate the square root of the right side: 1.12994943 Break this problem into two subproblems by setting (z + -0.75) equal to 1.12994943 and -1.12994943.Subproblem 1
z + -0.75 = 1.12994943 Simplifying z + -0.75 = 1.12994943 Reorder the terms: -0.75 + z = 1.12994943 Solving -0.75 + z = 1.12994943 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 = 1.12994943 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 1.12994943 + 0.75 z = 1.12994943 + 0.75 Combine like terms: 1.12994943 + 0.75 = 1.87994943 z = 1.87994943 Simplifying z = 1.87994943Subproblem 2
z + -0.75 = -1.12994943 Simplifying z + -0.75 = -1.12994943 Reorder the terms: -0.75 + z = -1.12994943 Solving -0.75 + z = -1.12994943 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 = -1.12994943 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -1.12994943 + 0.75 z = -1.12994943 + 0.75 Combine like terms: -1.12994943 + 0.75 = -0.37994943 z = -0.37994943 Simplifying z = -0.37994943Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.87994943, -0.37994943}
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