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Simplifying 38z2 + 96z + 15 = 0 Reorder the terms: 15 + 96z + 38z2 = 0 Solving 15 + 96z + 38z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 38 the coefficient of the squared term: Divide each side by '38'. 0.3947368421 + 2.526315789z + z2 = 0 Move the constant term to the right: Add '-0.3947368421' to each side of the equation. 0.3947368421 + 2.526315789z + -0.3947368421 + z2 = 0 + -0.3947368421 Reorder the terms: 0.3947368421 + -0.3947368421 + 2.526315789z + z2 = 0 + -0.3947368421 Combine like terms: 0.3947368421 + -0.3947368421 = 0.0000000000 0.0000000000 + 2.526315789z + z2 = 0 + -0.3947368421 2.526315789z + z2 = 0 + -0.3947368421 Combine like terms: 0 + -0.3947368421 = -0.3947368421 2.526315789z + z2 = -0.3947368421 The z term is 2.526315789z. Take half its coefficient (1.263157895). Square it (1.595567868) and add it to both sides. Add '1.595567868' to each side of the equation. 2.526315789z + 1.595567868 + z2 = -0.3947368421 + 1.595567868 Reorder the terms: 1.595567868 + 2.526315789z + z2 = -0.3947368421 + 1.595567868 Combine like terms: -0.3947368421 + 1.595567868 = 1.2008310259 1.595567868 + 2.526315789z + z2 = 1.2008310259 Factor a perfect square on the left side: (z + 1.263157895)(z + 1.263157895) = 1.2008310259 Calculate the square root of the right side: 1.095824359 Break this problem into two subproblems by setting (z + 1.263157895) equal to 1.095824359 and -1.095824359.Subproblem 1
z + 1.263157895 = 1.095824359 Simplifying z + 1.263157895 = 1.095824359 Reorder the terms: 1.263157895 + z = 1.095824359 Solving 1.263157895 + z = 1.095824359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.263157895' to each side of the equation. 1.263157895 + -1.263157895 + z = 1.095824359 + -1.263157895 Combine like terms: 1.263157895 + -1.263157895 = 0.000000000 0.000000000 + z = 1.095824359 + -1.263157895 z = 1.095824359 + -1.263157895 Combine like terms: 1.095824359 + -1.263157895 = -0.167333536 z = -0.167333536 Simplifying z = -0.167333536Subproblem 2
z + 1.263157895 = -1.095824359 Simplifying z + 1.263157895 = -1.095824359 Reorder the terms: 1.263157895 + z = -1.095824359 Solving 1.263157895 + z = -1.095824359 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.263157895' to each side of the equation. 1.263157895 + -1.263157895 + z = -1.095824359 + -1.263157895 Combine like terms: 1.263157895 + -1.263157895 = 0.000000000 0.000000000 + z = -1.095824359 + -1.263157895 z = -1.095824359 + -1.263157895 Combine like terms: -1.095824359 + -1.263157895 = -2.358982254 z = -2.358982254 Simplifying z = -2.358982254Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.167333536, -2.358982254}
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