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Simplifying -24z5 + 26z4 + 18z3 = 0 Reorder the terms: 18z3 + 26z4 + -24z5 = 0 Solving 18z3 + 26z4 + -24z5 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2z3'. 2z3(9 + 13z + -12z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'z3' equal to zero and attempt to solve: Simplifying z3 = 0 Solving z3 = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z3 = 0 The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.Subproblem 2
Set the factor '(9 + 13z + -12z2)' equal to zero and attempt to solve: Simplifying 9 + 13z + -12z2 = 0 Solving 9 + 13z + -12z2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -0.75 + -1.083333333z + z2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + -1.083333333z + 0.75 + z2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + -1.083333333z + z2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + -1.083333333z + z2 = 0 + 0.75 -1.083333333z + z2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 -1.083333333z + z2 = 0.75 The z term is -1.083333333z. Take half its coefficient (-0.5416666665). Square it (0.2934027776) and add it to both sides. Add '0.2934027776' to each side of the equation. -1.083333333z + 0.2934027776 + z2 = 0.75 + 0.2934027776 Reorder the terms: 0.2934027776 + -1.083333333z + z2 = 0.75 + 0.2934027776 Combine like terms: 0.75 + 0.2934027776 = 1.0434027776 0.2934027776 + -1.083333333z + z2 = 1.0434027776 Factor a perfect square on the left side: (z + -0.5416666665)(z + -0.5416666665) = 1.0434027776 Calculate the square root of the right side: 1.021470889 Break this problem into two subproblems by setting (z + -0.5416666665) equal to 1.021470889 and -1.021470889.Subproblem 1
z + -0.5416666665 = 1.021470889 Simplifying z + -0.5416666665 = 1.021470889 Reorder the terms: -0.5416666665 + z = 1.021470889 Solving -0.5416666665 + z = 1.021470889 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + z = 1.021470889 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + z = 1.021470889 + 0.5416666665 z = 1.021470889 + 0.5416666665 Combine like terms: 1.021470889 + 0.5416666665 = 1.5631375555 z = 1.5631375555 Simplifying z = 1.5631375555Subproblem 2
z + -0.5416666665 = -1.021470889 Simplifying z + -0.5416666665 = -1.021470889 Reorder the terms: -0.5416666665 + z = -1.021470889 Solving -0.5416666665 + z = -1.021470889 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.5416666665' to each side of the equation. -0.5416666665 + 0.5416666665 + z = -1.021470889 + 0.5416666665 Combine like terms: -0.5416666665 + 0.5416666665 = 0.0000000000 0.0000000000 + z = -1.021470889 + 0.5416666665 z = -1.021470889 + 0.5416666665 Combine like terms: -1.021470889 + 0.5416666665 = -0.4798042225 z = -0.4798042225 Simplifying z = -0.4798042225Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.5631375555, -0.4798042225}Solution
z = {1.5631375555, -0.4798042225}
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