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Simplifying 6z5 + -30z4 + 15z3 = 0 Reorder the terms: 15z3 + -30z4 + 6z5 = 0 Solving 15z3 + -30z4 + 6z5 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '3z3'. 3z3(5 + -10z + 2z2) = 0 Ignore the factor 3.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 '(5 + -10z + 2z2)' equal to zero and attempt to solve: Simplifying 5 + -10z + 2z2 = 0 Solving 5 + -10z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 2.5 + -5z + z2 = 0 Move the constant term to the right: Add '-2.5' to each side of the equation. 2.5 + -5z + -2.5 + z2 = 0 + -2.5 Reorder the terms: 2.5 + -2.5 + -5z + z2 = 0 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + -5z + z2 = 0 + -2.5 -5z + z2 = 0 + -2.5 Combine like terms: 0 + -2.5 = -2.5 -5z + z2 = -2.5 The z term is -5z. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5z + 6.25 + z2 = -2.5 + 6.25 Reorder the terms: 6.25 + -5z + z2 = -2.5 + 6.25 Combine like terms: -2.5 + 6.25 = 3.75 6.25 + -5z + z2 = 3.75 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 3.75 Calculate the square root of the right side: 1.936491673 Break this problem into two subproblems by setting (z + -2.5) equal to 1.936491673 and -1.936491673.Subproblem 1
z + -2.5 = 1.936491673 Simplifying z + -2.5 = 1.936491673 Reorder the terms: -2.5 + z = 1.936491673 Solving -2.5 + z = 1.936491673 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = 1.936491673 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 1.936491673 + 2.5 z = 1.936491673 + 2.5 Combine like terms: 1.936491673 + 2.5 = 4.436491673 z = 4.436491673 Simplifying z = 4.436491673Subproblem 2
z + -2.5 = -1.936491673 Simplifying z + -2.5 = -1.936491673 Reorder the terms: -2.5 + z = -1.936491673 Solving -2.5 + z = -1.936491673 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + z = -1.936491673 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -1.936491673 + 2.5 z = -1.936491673 + 2.5 Combine like terms: -1.936491673 + 2.5 = 0.563508327 z = 0.563508327 Simplifying z = 0.563508327Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.436491673, 0.563508327}Solution
z = {4.436491673, 0.563508327}
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