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Simplifying 16z2 + -96z + 142 = 0 Reorder the terms: 142 + -96z + 16z2 = 0 Solving 142 + -96z + 16z2 = 0 Solving for variable 'z'. Factor out the Greatest Common Factor (GCF), '2'. 2(71 + -48z + 8z2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(71 + -48z + 8z2)' equal to zero and attempt to solve: Simplifying 71 + -48z + 8z2 = 0 Solving 71 + -48z + 8z2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 8.875 + -6z + z2 = 0 Move the constant term to the right: Add '-8.875' to each side of the equation. 8.875 + -6z + -8.875 + z2 = 0 + -8.875 Reorder the terms: 8.875 + -8.875 + -6z + z2 = 0 + -8.875 Combine like terms: 8.875 + -8.875 = 0.000 0.000 + -6z + z2 = 0 + -8.875 -6z + z2 = 0 + -8.875 Combine like terms: 0 + -8.875 = -8.875 -6z + z2 = -8.875 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 = -8.875 + 9 Reorder the terms: 9 + -6z + z2 = -8.875 + 9 Combine like terms: -8.875 + 9 = 0.125 9 + -6z + z2 = 0.125 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.125 Calculate the square root of the right side: 0.353553391 Break this problem into two subproblems by setting (z + -3) equal to 0.353553391 and -0.353553391.Subproblem 1
z + -3 = 0.353553391 Simplifying z + -3 = 0.353553391 Reorder the terms: -3 + z = 0.353553391 Solving -3 + z = 0.353553391 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 = 0.353553391 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.353553391 + 3 z = 0.353553391 + 3 Combine like terms: 0.353553391 + 3 = 3.353553391 z = 3.353553391 Simplifying z = 3.353553391Subproblem 2
z + -3 = -0.353553391 Simplifying z + -3 = -0.353553391 Reorder the terms: -3 + z = -0.353553391 Solving -3 + z = -0.353553391 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 = -0.353553391 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.353553391 + 3 z = -0.353553391 + 3 Combine like terms: -0.353553391 + 3 = 2.646446609 z = 2.646446609 Simplifying z = 2.646446609Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.353553391, 2.646446609}Solution
z = {3.353553391, 2.646446609}
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