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Simplifying 28z5 + 21z4 + -56z3 = 0 Reorder the terms: -56z3 + 21z4 + 28z5 = 0 Solving -56z3 + 21z4 + 28z5 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '7z3'. 7z3(-8 + 3z + 4z2) = 0 Ignore the factor 7.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 '(-8 + 3z + 4z2)' equal to zero and attempt to solve: Simplifying -8 + 3z + 4z2 = 0 Solving -8 + 3z + 4z2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2 + 0.75z + z2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.75z + 2 + z2 = 0 + 2 Reorder the terms: -2 + 2 + 0.75z + z2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.75z + z2 = 0 + 2 0.75z + z2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.75z + z2 = 2 The z term is 0.75z. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75z + 0.140625 + z2 = 2 + 0.140625 Reorder the terms: 0.140625 + 0.75z + z2 = 2 + 0.140625 Combine like terms: 2 + 0.140625 = 2.140625 0.140625 + 0.75z + z2 = 2.140625 Factor a perfect square on the left side: (z + 0.375)(z + 0.375) = 2.140625 Calculate the square root of the right side: 1.463087489 Break this problem into two subproblems by setting (z + 0.375) equal to 1.463087489 and -1.463087489.Subproblem 1
z + 0.375 = 1.463087489 Simplifying z + 0.375 = 1.463087489 Reorder the terms: 0.375 + z = 1.463087489 Solving 0.375 + z = 1.463087489 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = 1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = 1.463087489 + -0.375 z = 1.463087489 + -0.375 Combine like terms: 1.463087489 + -0.375 = 1.088087489 z = 1.088087489 Simplifying z = 1.088087489Subproblem 2
z + 0.375 = -1.463087489 Simplifying z + 0.375 = -1.463087489 Reorder the terms: 0.375 + z = -1.463087489 Solving 0.375 + z = -1.463087489 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + z = -1.463087489 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + z = -1.463087489 + -0.375 z = -1.463087489 + -0.375 Combine like terms: -1.463087489 + -0.375 = -1.838087489 z = -1.838087489 Simplifying z = -1.838087489Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.088087489, -1.838087489}Solution
z = {1.088087489, -1.838087489}
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