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Simplifying 0 = 9z2 + 6z + -135 Reorder the terms: 0 = -135 + 6z + 9z2 Solving 0 = -135 + 6z + 9z2 Solving for variable 'z'. Combine like terms: 0 + 135 = 135 135 + -6z + -9z2 = -135 + 6z + 9z2 + 135 + -6z + -9z2 Reorder the terms: 135 + -6z + -9z2 = -135 + 135 + 6z + -6z + 9z2 + -9z2 Combine like terms: -135 + 135 = 0 135 + -6z + -9z2 = 0 + 6z + -6z + 9z2 + -9z2 135 + -6z + -9z2 = 6z + -6z + 9z2 + -9z2 Combine like terms: 6z + -6z = 0 135 + -6z + -9z2 = 0 + 9z2 + -9z2 135 + -6z + -9z2 = 9z2 + -9z2 Combine like terms: 9z2 + -9z2 = 0 135 + -6z + -9z2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(45 + -2z + -3z2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(45 + -2z + -3z2)' equal to zero and attempt to solve: Simplifying 45 + -2z + -3z2 = 0 Solving 45 + -2z + -3z2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -15 + 0.6666666667z + z2 = 0 Move the constant term to the right: Add '15' to each side of the equation. -15 + 0.6666666667z + 15 + z2 = 0 + 15 Reorder the terms: -15 + 15 + 0.6666666667z + z2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 0.6666666667z + z2 = 0 + 15 0.6666666667z + z2 = 0 + 15 Combine like terms: 0 + 15 = 15 0.6666666667z + z2 = 15 The z term is 0.6666666667z. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667z + 0.1111111112 + z2 = 15 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667z + z2 = 15 + 0.1111111112 Combine like terms: 15 + 0.1111111112 = 15.1111111112 0.1111111112 + 0.6666666667z + z2 = 15.1111111112 Factor a perfect square on the left side: (z + 0.3333333334)(z + 0.3333333334) = 15.1111111112 Calculate the square root of the right side: 3.887301263 Break this problem into two subproblems by setting (z + 0.3333333334) equal to 3.887301263 and -3.887301263.Subproblem 1
z + 0.3333333334 = 3.887301263 Simplifying z + 0.3333333334 = 3.887301263 Reorder the terms: 0.3333333334 + z = 3.887301263 Solving 0.3333333334 + z = 3.887301263 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = 3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = 3.887301263 + -0.3333333334 z = 3.887301263 + -0.3333333334 Combine like terms: 3.887301263 + -0.3333333334 = 3.5539679296 z = 3.5539679296 Simplifying z = 3.5539679296Subproblem 2
z + 0.3333333334 = -3.887301263 Simplifying z + 0.3333333334 = -3.887301263 Reorder the terms: 0.3333333334 + z = -3.887301263 Solving 0.3333333334 + z = -3.887301263 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + z = -3.887301263 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + z = -3.887301263 + -0.3333333334 z = -3.887301263 + -0.3333333334 Combine like terms: -3.887301263 + -0.3333333334 = -4.2206345964 z = -4.2206345964 Simplifying z = -4.2206345964Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.5539679296, -4.2206345964}Solution
z = {3.5539679296, -4.2206345964}
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