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Simplifying -3z2 + 72z = 15 Reorder the terms: 72z + -3z2 = 15 Solving 72z + -3z2 = 15 Solving for variable 'z'. Reorder the terms: -15 + 72z + -3z2 = 15 + -15 Combine like terms: 15 + -15 = 0 -15 + 72z + -3z2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-5 + 24z + -1z2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-5 + 24z + -1z2)' equal to zero and attempt to solve: Simplifying -5 + 24z + -1z2 = 0 Solving -5 + 24z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 5 + -24z + z2 = 0 Move the constant term to the right: Add '-5' to each side of the equation. 5 + -24z + -5 + z2 = 0 + -5 Reorder the terms: 5 + -5 + -24z + z2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -24z + z2 = 0 + -5 -24z + z2 = 0 + -5 Combine like terms: 0 + -5 = -5 -24z + z2 = -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 = -5 + 144 Reorder the terms: 144 + -24z + z2 = -5 + 144 Combine like terms: -5 + 144 = 139 144 + -24z + z2 = 139 Factor a perfect square on the left side: (z + -12)(z + -12) = 139 Calculate the square root of the right side: 11.789826123 Break this problem into two subproblems by setting (z + -12) equal to 11.789826123 and -11.789826123.Subproblem 1
z + -12 = 11.789826123 Simplifying z + -12 = 11.789826123 Reorder the terms: -12 + z = 11.789826123 Solving -12 + z = 11.789826123 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 = 11.789826123 + 12 Combine like terms: -12 + 12 = 0 0 + z = 11.789826123 + 12 z = 11.789826123 + 12 Combine like terms: 11.789826123 + 12 = 23.789826123 z = 23.789826123 Simplifying z = 23.789826123Subproblem 2
z + -12 = -11.789826123 Simplifying z + -12 = -11.789826123 Reorder the terms: -12 + z = -11.789826123 Solving -12 + z = -11.789826123 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 = -11.789826123 + 12 Combine like terms: -12 + 12 = 0 0 + z = -11.789826123 + 12 z = -11.789826123 + 12 Combine like terms: -11.789826123 + 12 = 0.210173877 z = 0.210173877 Simplifying z = 0.210173877Solution
The solution to the problem is based on the solutions from the subproblems. z = {23.789826123, 0.210173877}Solution
z = {23.789826123, 0.210173877}
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