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Simplifying 4z2 + -24z + 33 = 0 Reorder the terms: 33 + -24z + 4z2 = 0 Solving 33 + -24z + 4z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 8.25 + -6z + z2 = 0 Move the constant term to the right: Add '-8.25' to each side of the equation. 8.25 + -6z + -8.25 + z2 = 0 + -8.25 Reorder the terms: 8.25 + -8.25 + -6z + z2 = 0 + -8.25 Combine like terms: 8.25 + -8.25 = 0.00 0.00 + -6z + z2 = 0 + -8.25 -6z + z2 = 0 + -8.25 Combine like terms: 0 + -8.25 = -8.25 -6z + z2 = -8.25 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 = -8.25 + 9 Reorder the terms: 9 + -6z + z2 = -8.25 + 9 Combine like terms: -8.25 + 9 = 0.75 9 + -6z + z2 = 0.75 Factor a perfect square on the left side: (z + -3)(z + -3) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (z + -3) equal to 0.866025404 and -0.866025404.Subproblem 1
z + -3 = 0.866025404 Simplifying z + -3 = 0.866025404 Reorder the terms: -3 + z = 0.866025404 Solving -3 + z = 0.866025404 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 = 0.866025404 + 3 Combine like terms: -3 + 3 = 0 0 + z = 0.866025404 + 3 z = 0.866025404 + 3 Combine like terms: 0.866025404 + 3 = 3.866025404 z = 3.866025404 Simplifying z = 3.866025404Subproblem 2
z + -3 = -0.866025404 Simplifying z + -3 = -0.866025404 Reorder the terms: -3 + z = -0.866025404 Solving -3 + z = -0.866025404 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 = -0.866025404 + 3 Combine like terms: -3 + 3 = 0 0 + z = -0.866025404 + 3 z = -0.866025404 + 3 Combine like terms: -0.866025404 + 3 = 2.133974596 z = 2.133974596 Simplifying z = 2.133974596Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.866025404, 2.133974596}
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