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Simplifying -36z2 + 65z + 30 = 0 Reorder the terms: 30 + 65z + -36z2 = 0 Solving 30 + 65z + -36z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -36 the coefficient of the squared term: Divide each side by '-36'. -0.8333333333 + -1.805555556z + z2 = 0 Move the constant term to the right: Add '0.8333333333' to each side of the equation. -0.8333333333 + -1.805555556z + 0.8333333333 + z2 = 0 + 0.8333333333 Reorder the terms: -0.8333333333 + 0.8333333333 + -1.805555556z + z2 = 0 + 0.8333333333 Combine like terms: -0.8333333333 + 0.8333333333 = 0.0000000000 0.0000000000 + -1.805555556z + z2 = 0 + 0.8333333333 -1.805555556z + z2 = 0 + 0.8333333333 Combine like terms: 0 + 0.8333333333 = 0.8333333333 -1.805555556z + z2 = 0.8333333333 The z term is -1.805555556z. Take half its coefficient (-0.902777778). Square it (0.8150077165) and add it to both sides. Add '0.8150077165' to each side of the equation. -1.805555556z + 0.8150077165 + z2 = 0.8333333333 + 0.8150077165 Reorder the terms: 0.8150077165 + -1.805555556z + z2 = 0.8333333333 + 0.8150077165 Combine like terms: 0.8333333333 + 0.8150077165 = 1.6483410498 0.8150077165 + -1.805555556z + z2 = 1.6483410498 Factor a perfect square on the left side: (z + -0.902777778)(z + -0.902777778) = 1.6483410498 Calculate the square root of the right side: 1.28387735 Break this problem into two subproblems by setting (z + -0.902777778) equal to 1.28387735 and -1.28387735.Subproblem 1
z + -0.902777778 = 1.28387735 Simplifying z + -0.902777778 = 1.28387735 Reorder the terms: -0.902777778 + z = 1.28387735 Solving -0.902777778 + z = 1.28387735 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.902777778' to each side of the equation. -0.902777778 + 0.902777778 + z = 1.28387735 + 0.902777778 Combine like terms: -0.902777778 + 0.902777778 = 0.000000000 0.000000000 + z = 1.28387735 + 0.902777778 z = 1.28387735 + 0.902777778 Combine like terms: 1.28387735 + 0.902777778 = 2.186655128 z = 2.186655128 Simplifying z = 2.186655128Subproblem 2
z + -0.902777778 = -1.28387735 Simplifying z + -0.902777778 = -1.28387735 Reorder the terms: -0.902777778 + z = -1.28387735 Solving -0.902777778 + z = -1.28387735 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.902777778' to each side of the equation. -0.902777778 + 0.902777778 + z = -1.28387735 + 0.902777778 Combine like terms: -0.902777778 + 0.902777778 = 0.000000000 0.000000000 + z = -1.28387735 + 0.902777778 z = -1.28387735 + 0.902777778 Combine like terms: -1.28387735 + 0.902777778 = -0.381099572 z = -0.381099572 Simplifying z = -0.381099572Solution
The solution to the problem is based on the solutions from the subproblems. z = {2.186655128, -0.381099572}
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