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Simplifying 4z2 + -24z + 30 = 0 Reorder the terms: 30 + -24z + 4z2 = 0 Solving 30 + -24z + 4z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(15 + -12z + 2z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(15 + -12z + 2z2)' equal to zero and attempt to solve: Simplifying 15 + -12z + 2z2 = 0 Solving 15 + -12z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 7.5 + -6z + z2 = 0 Move the constant term to the right: Add '-7.5' to each side of the equation. 7.5 + -6z + -7.5 + z2 = 0 + -7.5 Reorder the terms: 7.5 + -7.5 + -6z + z2 = 0 + -7.5 Combine like terms: 7.5 + -7.5 = 0.0 0.0 + -6z + z2 = 0 + -7.5 -6z + z2 = 0 + -7.5 Combine like terms: 0 + -7.5 = -7.5 -6z + z2 = -7.5 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -7.5 + 9 Reorder the terms: 9 + -6z + z2 = -7.5 + 9 Combine like terms: -7.5 + 9 = 1.5 9 + -6z + z2 = 1.5 Factor a perfect square on the left side: (z + -3)(z + -3) = 1.5 Calculate the square root of the right side: 1.224744871 Break this problem into two subproblems by setting (z + -3) equal to 1.224744871 and -1.224744871.Subproblem 1
z + -3 = 1.224744871 Simplifying z + -3 = 1.224744871 Reorder the terms: -3 + z = 1.224744871 Solving -3 + z = 1.224744871 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 1.224744871 + 3 Combine like terms: -3 + 3 = 0 0 + z = 1.224744871 + 3 z = 1.224744871 + 3 Combine like terms: 1.224744871 + 3 = 4.224744871 z = 4.224744871 Simplifying z = 4.224744871Subproblem 2
z + -3 = -1.224744871 Simplifying z + -3 = -1.224744871 Reorder the terms: -3 + z = -1.224744871 Solving -3 + z = -1.224744871 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -1.224744871 + 3 Combine like terms: -3 + 3 = 0 0 + z = -1.224744871 + 3 z = -1.224744871 + 3 Combine like terms: -1.224744871 + 3 = 1.775255129 z = 1.775255129 Simplifying z = 1.775255129Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.224744871, 1.775255129}Solution
z = {4.224744871, 1.775255129}
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