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Simplifying -4z2 + 96z + 18 = 0 Reorder the terms: 18 + 96z + -4z2 = 0 Solving 18 + 96z + -4z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(9 + 48z + -2z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(9 + 48z + -2z2)' equal to zero and attempt to solve: Simplifying 9 + 48z + -2z2 = 0 Solving 9 + 48z + -2z2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -4.5 + -24z + z2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + -24z + 4.5 + z2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + -24z + z2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + -24z + z2 = 0 + 4.5 -24z + z2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 -24z + z2 = 4.5 The z term is -24z. Take half its coefficient (-12). Square it (144) and add it to both sides. Add '144' to each side of the equation. -24z + 144 + z2 = 4.5 + 144 Reorder the terms: 144 + -24z + z2 = 4.5 + 144 Combine like terms: 4.5 + 144 = 148.5 144 + -24z + z2 = 148.5 Factor a perfect square on the left side: (z + -12)(z + -12) = 148.5 Calculate the square root of the right side: 12.186057607 Break this problem into two subproblems by setting (z + -12) equal to 12.186057607 and -12.186057607.Subproblem 1
z + -12 = 12.186057607 Simplifying z + -12 = 12.186057607 Reorder the terms: -12 + z = 12.186057607 Solving -12 + z = 12.186057607 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = 12.186057607 + 12 Combine like terms: -12 + 12 = 0 0 + z = 12.186057607 + 12 z = 12.186057607 + 12 Combine like terms: 12.186057607 + 12 = 24.186057607 z = 24.186057607 Simplifying z = 24.186057607Subproblem 2
z + -12 = -12.186057607 Simplifying z + -12 = -12.186057607 Reorder the terms: -12 + z = -12.186057607 Solving -12 + z = -12.186057607 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '12' to each side of the equation. -12 + 12 + z = -12.186057607 + 12 Combine like terms: -12 + 12 = 0 0 + z = -12.186057607 + 12 z = -12.186057607 + 12 Combine like terms: -12.186057607 + 12 = -0.186057607 z = -0.186057607 Simplifying z = -0.186057607Solution
The solution to the problem is based on the solutions from the subproblems. z = {24.186057607, -0.186057607}Solution
z = {24.186057607, -0.186057607}
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