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Simplifying 14z4 + -70z3 + 10z2 = 0 Reorder the terms: 10z2 + -70z3 + 14z4 = 0 Solving 10z2 + -70z3 + 14z4 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2z2'. 2z2(5 + -35z + 7z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve: Simplifying z2 = 0 Solving z2 = 0 Move all terms containing z to the left, all other terms to the right. Simplifying z2 = 0 Take the square root of each side: z = {0}Subproblem 2
Set the factor '(5 + -35z + 7z2)' equal to zero and attempt to solve: Simplifying 5 + -35z + 7z2 = 0 Solving 5 + -35z + 7z2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.7142857143 + -5z + z2 = 0 Move the constant term to the right: Add '-0.7142857143' to each side of the equation. 0.7142857143 + -5z + -0.7142857143 + z2 = 0 + -0.7142857143 Reorder the terms: 0.7142857143 + -0.7142857143 + -5z + z2 = 0 + -0.7142857143 Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000 0.0000000000 + -5z + z2 = 0 + -0.7142857143 -5z + z2 = 0 + -0.7142857143 Combine like terms: 0 + -0.7142857143 = -0.7142857143 -5z + z2 = -0.7142857143 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 = -0.7142857143 + 6.25 Reorder the terms: 6.25 + -5z + z2 = -0.7142857143 + 6.25 Combine like terms: -0.7142857143 + 6.25 = 5.5357142857 6.25 + -5z + z2 = 5.5357142857 Factor a perfect square on the left side: (z + -2.5)(z + -2.5) = 5.5357142857 Calculate the square root of the right side: 2.35280987 Break this problem into two subproblems by setting (z + -2.5) equal to 2.35280987 and -2.35280987.Subproblem 1
z + -2.5 = 2.35280987 Simplifying z + -2.5 = 2.35280987 Reorder the terms: -2.5 + z = 2.35280987 Solving -2.5 + z = 2.35280987 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 = 2.35280987 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = 2.35280987 + 2.5 z = 2.35280987 + 2.5 Combine like terms: 2.35280987 + 2.5 = 4.85280987 z = 4.85280987 Simplifying z = 4.85280987Subproblem 2
z + -2.5 = -2.35280987 Simplifying z + -2.5 = -2.35280987 Reorder the terms: -2.5 + z = -2.35280987 Solving -2.5 + z = -2.35280987 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 = -2.35280987 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + z = -2.35280987 + 2.5 z = -2.35280987 + 2.5 Combine like terms: -2.35280987 + 2.5 = 0.14719013 z = 0.14719013 Simplifying z = 0.14719013Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.85280987, 0.14719013}Solution
z = {0, 4.85280987, 0.14719013}
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