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Simplifying 4z(2z + -3) = -2 + 3 Reorder the terms: 4z(-3 + 2z) = -2 + 3 (-3 * 4z + 2z * 4z) = -2 + 3 (-12z + 8z2) = -2 + 3 Combine like terms: -2 + 3 = 1 -12z + 8z2 = 1 Solving -12z + 8z2 = 1 Solving for variable 'z'. Reorder the terms: -1 + -12z + 8z2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + -12z + 8z2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -1.5z + z2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -1.5z + 0.125 + z2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -1.5z + z2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -1.5z + z2 = 0 + 0.125 -1.5z + z2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -1.5z + z2 = 0.125 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 = 0.125 + 0.5625 Reorder the terms: 0.5625 + -1.5z + z2 = 0.125 + 0.5625 Combine like terms: 0.125 + 0.5625 = 0.6875 0.5625 + -1.5z + z2 = 0.6875 Factor a perfect square on the left side: (z + -0.75)(z + -0.75) = 0.6875 Calculate the square root of the right side: 0.829156198 Break this problem into two subproblems by setting (z + -0.75) equal to 0.829156198 and -0.829156198.Subproblem 1
z + -0.75 = 0.829156198 Simplifying z + -0.75 = 0.829156198 Reorder the terms: -0.75 + z = 0.829156198 Solving -0.75 + z = 0.829156198 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 = 0.829156198 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = 0.829156198 + 0.75 z = 0.829156198 + 0.75 Combine like terms: 0.829156198 + 0.75 = 1.579156198 z = 1.579156198 Simplifying z = 1.579156198Subproblem 2
z + -0.75 = -0.829156198 Simplifying z + -0.75 = -0.829156198 Reorder the terms: -0.75 + z = -0.829156198 Solving -0.75 + z = -0.829156198 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 = -0.829156198 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + z = -0.829156198 + 0.75 z = -0.829156198 + 0.75 Combine like terms: -0.829156198 + 0.75 = -0.079156198 z = -0.079156198 Simplifying z = -0.079156198Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.579156198, -0.079156198}
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