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Simplifying 16z2 + 137 = 96z Reorder the terms: 137 + 16z2 = 96z Solving 137 + 16z2 = 96z Solving for variable 'z'. Reorder the terms: 137 + -96z + 16z2 = 96z + -96z Combine like terms: 96z + -96z = 0 137 + -96z + 16z2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. 8.5625 + -6z + z2 = 0 Move the constant term to the right: Add '-8.5625' to each side of the equation. 8.5625 + -6z + -8.5625 + z2 = 0 + -8.5625 Reorder the terms: 8.5625 + -8.5625 + -6z + z2 = 0 + -8.5625 Combine like terms: 8.5625 + -8.5625 = 0.0000 0.0000 + -6z + z2 = 0 + -8.5625 -6z + z2 = 0 + -8.5625 Combine like terms: 0 + -8.5625 = -8.5625 -6z + z2 = -8.5625 The z term is -6z. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6z + 9 + z2 = -8.5625 + 9 Reorder the terms: 9 + -6z + z2 = -8.5625 + 9 Combine like terms: -8.5625 + 9 = 0.4375 9 + -6z + z2 = 0.4375 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (z + -3) equal to 0.661437828 and -0.661437828.Subproblem 1
z + -3 = 0.661437828 Simplifying z + -3 = 0.661437828 Reorder the terms: -3 + z = 0.661437828 Solving -3 + z = 0.661437828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = 0.661437828 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.661437828 + 3 z = 0.661437828 + 3 Combine like terms: 0.661437828 + 3 = 3.661437828 z = 3.661437828 Simplifying z = 3.661437828Subproblem 2
z + -3 = -0.661437828 Simplifying z + -3 = -0.661437828 Reorder the terms: -3 + z = -0.661437828 Solving -3 + z = -0.661437828 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + z = -0.661437828 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.661437828 + 3 z = -0.661437828 + 3 Combine like terms: -0.661437828 + 3 = 2.338562172 z = 2.338562172 Simplifying z = 2.338562172Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.661437828, 2.338562172}
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