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Simplifying 5z(4z + 2) = 166 Reorder the terms: 5z(2 + 4z) = 166 (2 * 5z + 4z * 5z) = 166 (10z + 20z2) = 166 Solving 10z + 20z2 = 166 Solving for variable 'z'. Reorder the terms: -166 + 10z + 20z2 = 166 + -166 Combine like terms: 166 + -166 = 0 -166 + 10z + 20z2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-83 + 5z + 10z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-83 + 5z + 10z2)' equal to zero and attempt to solve: Simplifying -83 + 5z + 10z2 = 0 Solving -83 + 5z + 10z2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -8.3 + 0.5z + z2 = 0.0 Move the constant term to the right: Add '8.3' to each side of the equation. -8.3 + 0.5z + 8.3 + z2 = 0.0 + 8.3 Reorder the terms: -8.3 + 8.3 + 0.5z + z2 = 0.0 + 8.3 Combine like terms: -8.3 + 8.3 = 0.0 0.0 + 0.5z + z2 = 0.0 + 8.3 0.5z + z2 = 0.0 + 8.3 Combine like terms: 0.0 + 8.3 = 8.3 0.5z + z2 = 8.3 The z term is 0.5z. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5z + 0.0625 + z2 = 8.3 + 0.0625 Reorder the terms: 0.0625 + 0.5z + z2 = 8.3 + 0.0625 Combine like terms: 8.3 + 0.0625 = 8.3625 0.0625 + 0.5z + z2 = 8.3625 Factor a perfect square on the left side: (z + 0.25)(z + 0.25) = 8.3625 Calculate the square root of the right side: 2.891798748 Break this problem into two subproblems by setting (z + 0.25) equal to 2.891798748 and -2.891798748.Subproblem 1
z + 0.25 = 2.891798748 Simplifying z + 0.25 = 2.891798748 Reorder the terms: 0.25 + z = 2.891798748 Solving 0.25 + z = 2.891798748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = 2.891798748 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = 2.891798748 + -0.25 z = 2.891798748 + -0.25 Combine like terms: 2.891798748 + -0.25 = 2.641798748 z = 2.641798748 Simplifying z = 2.641798748Subproblem 2
z + 0.25 = -2.891798748 Simplifying z + 0.25 = -2.891798748 Reorder the terms: 0.25 + z = -2.891798748 Solving 0.25 + z = -2.891798748 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + z = -2.891798748 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + z = -2.891798748 + -0.25 z = -2.891798748 + -0.25 Combine like terms: -2.891798748 + -0.25 = -3.141798748 z = -3.141798748 Simplifying z = -3.141798748Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.641798748, -3.141798748}Solution
z = {2.641798748, -3.141798748}
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