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Simplifying 2z2 + -3z = 7 Reorder the terms: -3z + 2z2 = 7 Solving -3z + 2z2 = 7 Solving for variable 'z'. Reorder the terms: -7 + -3z + 2z2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + -3z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -1.5z + z2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -1.5z + 3.5 + z2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -1.5z + z2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -1.5z + z2 = 0 + 3.5 -1.5z + z2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -1.5z + z2 = 3.5 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 = 3.5 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 3.5 + 0.5625 Combine like terms: 3.5 + 0.5625 = 4.0625 0.5625 + -1.5z + z2 = 4.0625 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (z + -0.75) equal to 2.015564437 and -2.015564437.Subproblem 1
z + -0.75 = 2.015564437 Simplifying z + -0.75 = 2.015564437 Reorder the terms: -0.75 + z = 2.015564437 Solving -0.75 + z = 2.015564437 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 = 2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 2.015564437 + 0.75 z = 2.015564437 + 0.75 Combine like terms: 2.015564437 + 0.75 = 2.765564437 z = 2.765564437 Simplifying z = 2.765564437Subproblem 2
z + -0.75 = -2.015564437 Simplifying z + -0.75 = -2.015564437 Reorder the terms: -0.75 + z = -2.015564437 Solving -0.75 + z = -2.015564437 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 = -2.015564437 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -2.015564437 + 0.75 z = -2.015564437 + 0.75 Combine like terms: -2.015564437 + 0.75 = -1.265564437 z = -1.265564437 Simplifying z = -1.265564437Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.765564437, -1.265564437}
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