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Simplifying 6z4 + -42z3 + 14z2 = 0 Reorder the terms: 14z2 + -42z3 + 6z4 = 0 Solving 14z2 + -42z3 + 6z4 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2z2'. 2z2(7 + -21z + 3z2) = 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 '(7 + -21z + 3z2)' equal to zero and attempt to solve: Simplifying 7 + -21z + 3z2 = 0 Solving 7 + -21z + 3z2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 2.333333333 + -7z + z2 = 0 Move the constant term to the right: Add '-2.333333333' to each side of the equation. 2.333333333 + -7z + -2.333333333 + z2 = 0 + -2.333333333 Reorder the terms: 2.333333333 + -2.333333333 + -7z + z2 = 0 + -2.333333333 Combine like terms: 2.333333333 + -2.333333333 = 0.000000000 0.000000000 + -7z + z2 = 0 + -2.333333333 -7z + z2 = 0 + -2.333333333 Combine like terms: 0 + -2.333333333 = -2.333333333 -7z + z2 = -2.333333333 The z term is -7z. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7z + 12.25 + z2 = -2.333333333 + 12.25 Reorder the terms: 12.25 + -7z + z2 = -2.333333333 + 12.25 Combine like terms: -2.333333333 + 12.25 = 9.916666667 12.25 + -7z + z2 = 9.916666667 Factor a perfect square on the left side: (z + -3.5)(z + -3.5) = 9.916666667 Calculate the square root of the right side: 3.149073938 Break this problem into two subproblems by setting (z + -3.5) equal to 3.149073938 and -3.149073938.Subproblem 1
z + -3.5 = 3.149073938 Simplifying z + -3.5 = 3.149073938 Reorder the terms: -3.5 + z = 3.149073938 Solving -3.5 + z = 3.149073938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + z = 3.149073938 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = 3.149073938 + 3.5 z = 3.149073938 + 3.5 Combine like terms: 3.149073938 + 3.5 = 6.649073938 z = 6.649073938 Simplifying z = 6.649073938Subproblem 2
z + -3.5 = -3.149073938 Simplifying z + -3.5 = -3.149073938 Reorder the terms: -3.5 + z = -3.149073938 Solving -3.5 + z = -3.149073938 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + z = -3.149073938 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + z = -3.149073938 + 3.5 z = -3.149073938 + 3.5 Combine like terms: -3.149073938 + 3.5 = 0.350926062 z = 0.350926062 Simplifying z = 0.350926062Solution
The solution to the problem is based on the solutions from the subproblems. z = {6.649073938, 0.350926062}Solution
z = {0, 6.649073938, 0.350926062}
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