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Simplifying -4z2 + 112z + 68 = 0 Reorder the terms: 68 + 112z + -4z2 = 0 Solving 68 + 112z + -4z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '4'. 4(17 + 28z + -1z2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(17 + 28z + -1z2)' equal to zero and attempt to solve: Simplifying 17 + 28z + -1z2 = 0 Solving 17 + 28z + -1z2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -17 + -28z + z2 = 0 Move the constant term to the right: Add '17' to each side of the equation. -17 + -28z + 17 + z2 = 0 + 17 Reorder the terms: -17 + 17 + -28z + z2 = 0 + 17 Combine like terms: -17 + 17 = 0 0 + -28z + z2 = 0 + 17 -28z + z2 = 0 + 17 Combine like terms: 0 + 17 = 17 -28z + z2 = 17 The z term is -28z. Take half its coefficient (-14). Square it (196) and add it to both sides. Add '196' to each side of the equation. -28z + 196 + z2 = 17 + 196 Reorder the terms: 196 + -28z + z2 = 17 + 196 Combine like terms: 17 + 196 = 213 196 + -28z + z2 = 213 Factor a perfect square on the left side: (z + -14)(z + -14) = 213 Calculate the square root of the right side: 14.594519519 Break this problem into two subproblems by setting (z + -14) equal to 14.594519519 and -14.594519519.Subproblem 1
z + -14 = 14.594519519 Simplifying z + -14 = 14.594519519 Reorder the terms: -14 + z = 14.594519519 Solving -14 + z = 14.594519519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '14' to each side of the equation. -14 + 14 + z = 14.594519519 + 14 Combine like terms: -14 + 14 = 0 0 + z = 14.594519519 + 14 z = 14.594519519 + 14 Combine like terms: 14.594519519 + 14 = 28.594519519 z = 28.594519519 Simplifying z = 28.594519519Subproblem 2
z + -14 = -14.594519519 Simplifying z + -14 = -14.594519519 Reorder the terms: -14 + z = -14.594519519 Solving -14 + z = -14.594519519 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '14' to each side of the equation. -14 + 14 + z = -14.594519519 + 14 Combine like terms: -14 + 14 = 0 0 + z = -14.594519519 + 14 z = -14.594519519 + 14 Combine like terms: -14.594519519 + 14 = -0.594519519 z = -0.594519519 Simplifying z = -0.594519519Solution
The solution to the problem is based on the solutions from the subproblems. z = {28.594519519, -0.594519519}Solution
z = {28.594519519, -0.594519519}
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