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Simplifying 2z2 = 3 + -8z Solving 2z2 = 3 + -8z Solving for variable 'z'. Reorder the terms: -3 + 8z + 2z2 = 3 + -8z + -3 + 8z Reorder the terms: -3 + 8z + 2z2 = 3 + -3 + -8z + 8z Combine like terms: 3 + -3 = 0 -3 + 8z + 2z2 = 0 + -8z + 8z -3 + 8z + 2z2 = -8z + 8z Combine like terms: -8z + 8z = 0 -3 + 8z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -1.5 + 4z + z2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 4z + 1.5 + z2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 4z + z2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 4z + z2 = 0 + 1.5 4z + z2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 4z + z2 = 1.5 The z term is 4z. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4z + 4 + z2 = 1.5 + 4 Reorder the terms: 4 + 4z + z2 = 1.5 + 4 Combine like terms: 1.5 + 4 = 5.5 4 + 4z + z2 = 5.5 Factor a perfect square on the left side: (z + 2)(z + 2) = 5.5 Calculate the square root of the right side: 2.34520788 Break this problem into two subproblems by setting (z + 2) equal to 2.34520788 and -2.34520788.Subproblem 1
z + 2 = 2.34520788 Simplifying z + 2 = 2.34520788 Reorder the terms: 2 + z = 2.34520788 Solving 2 + z = 2.34520788 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 2.34520788 + -2 Combine like terms: 2 + -2 = 0 0 + z = 2.34520788 + -2 z = 2.34520788 + -2 Combine like terms: 2.34520788 + -2 = 0.34520788 z = 0.34520788 Simplifying z = 0.34520788Subproblem 2
z + 2 = -2.34520788 Simplifying z + 2 = -2.34520788 Reorder the terms: 2 + z = -2.34520788 Solving 2 + z = -2.34520788 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -2.34520788 + -2 Combine like terms: 2 + -2 = 0 0 + z = -2.34520788 + -2 z = -2.34520788 + -2 Combine like terms: -2.34520788 + -2 = -4.34520788 z = -4.34520788 Simplifying z = -4.34520788Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.34520788, -4.34520788}
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