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Simplifying 9z2 + 30z + -38 = 0 Reorder the terms: -38 + 30z + 9z2 = 0 Solving -38 + 30z + 9z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -4.222222222 + 3.333333333z + z2 = 0 Move the constant term to the right: Add '4.222222222' to each side of the equation. -4.222222222 + 3.333333333z + 4.222222222 + z2 = 0 + 4.222222222 Reorder the terms: -4.222222222 + 4.222222222 + 3.333333333z + z2 = 0 + 4.222222222 Combine like terms: -4.222222222 + 4.222222222 = 0.000000000 0.000000000 + 3.333333333z + z2 = 0 + 4.222222222 3.333333333z + z2 = 0 + 4.222222222 Combine like terms: 0 + 4.222222222 = 4.222222222 3.333333333z + z2 = 4.222222222 The z term is 3.333333333z. Take half its coefficient (1.666666667). Square it (2.777777779) and add it to both sides. Add '2.777777779' to each side of the equation. 3.333333333z + 2.777777779 + z2 = 4.222222222 + 2.777777779 Reorder the terms: 2.777777779 + 3.333333333z + z2 = 4.222222222 + 2.777777779 Combine like terms: 4.222222222 + 2.777777779 = 7.000000001 2.777777779 + 3.333333333z + z2 = 7.000000001 Factor a perfect square on the left side: (z + 1.666666667)(z + 1.666666667) = 7.000000001 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (z + 1.666666667) equal to 2.645751311 and -2.645751311.Subproblem 1
z + 1.666666667 = 2.645751311 Simplifying z + 1.666666667 = 2.645751311 Reorder the terms: 1.666666667 + z = 2.645751311 Solving 1.666666667 + z = 2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + z = 2.645751311 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + z = 2.645751311 + -1.666666667 z = 2.645751311 + -1.666666667 Combine like terms: 2.645751311 + -1.666666667 = 0.979084644 z = 0.979084644 Simplifying z = 0.979084644Subproblem 2
z + 1.666666667 = -2.645751311 Simplifying z + 1.666666667 = -2.645751311 Reorder the terms: 1.666666667 + z = -2.645751311 Solving 1.666666667 + z = -2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.666666667' to each side of the equation. 1.666666667 + -1.666666667 + z = -2.645751311 + -1.666666667 Combine like terms: 1.666666667 + -1.666666667 = 0.000000000 0.000000000 + z = -2.645751311 + -1.666666667 z = -2.645751311 + -1.666666667 Combine like terms: -2.645751311 + -1.666666667 = -4.312417978 z = -4.312417978 Simplifying z = -4.312417978Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.979084644, -4.312417978}
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