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Simplifying 2z2 + 3z + -6 = 0 Reorder the terms: -6 + 3z + 2z2 = 0 Solving -6 + 3z + 2z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3 + 1.5z + z2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 1.5z + 3 + z2 = 0 + 3 Reorder the terms: -3 + 3 + 1.5z + z2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 1.5z + z2 = 0 + 3 1.5z + z2 = 0 + 3 Combine like terms: 0 + 3 = 3 1.5z + z2 = 3 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 + 0.5625 Reorder the terms: 0.5625 + 1.5z + z2 = 3 + 0.5625 Combine like terms: 3 + 0.5625 = 3.5625 0.5625 + 1.5z + z2 = 3.5625 Factor a perfect square on the left side: (z + 0.75)(z + 0.75) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (z + 0.75) equal to 1.887458609 and -1.887458609.Subproblem 1
z + 0.75 = 1.887458609 Simplifying z + 0.75 = 1.887458609 Reorder the terms: 0.75 + z = 1.887458609 Solving 0.75 + z = 1.887458609 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 = 1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.887458609 + -0.75 z = 1.887458609 + -0.75 Combine like terms: 1.887458609 + -0.75 = 1.137458609 z = 1.137458609 Simplifying z = 1.137458609Subproblem 2
z + 0.75 = -1.887458609 Simplifying z + 0.75 = -1.887458609 Reorder the terms: 0.75 + z = -1.887458609 Solving 0.75 + z = -1.887458609 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 = -1.887458609 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.887458609 + -0.75 z = -1.887458609 + -0.75 Combine like terms: -1.887458609 + -0.75 = -2.637458609 z = -2.637458609 Simplifying z = -2.637458609Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.137458609, -2.637458609}
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