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Simplifying 33z2 + -3z = 54 Reorder the terms: -3z + 33z2 = 54 Solving -3z + 33z2 = 54 Solving for variable 'z'. Reorder the terms: -54 + -3z + 33z2 = 54 + -54 Combine like terms: 54 + -54 = 0 -54 + -3z + 33z2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-18 + -1z + 11z2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-18 + -1z + 11z2)' equal to zero and attempt to solve: Simplifying -18 + -1z + 11z2 = 0 Solving -18 + -1z + 11z2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.636363636 + -0.09090909091z + z2 = 0 Move the constant term to the right: Add '1.636363636' to each side of the equation. -1.636363636 + -0.09090909091z + 1.636363636 + z2 = 0 + 1.636363636 Reorder the terms: -1.636363636 + 1.636363636 + -0.09090909091z + z2 = 0 + 1.636363636 Combine like terms: -1.636363636 + 1.636363636 = 0.000000000 0.000000000 + -0.09090909091z + z2 = 0 + 1.636363636 -0.09090909091z + z2 = 0 + 1.636363636 Combine like terms: 0 + 1.636363636 = 1.636363636 -0.09090909091z + z2 = 1.636363636 The z term is -0.09090909091z. Take half its coefficient (-0.04545454546). Square it (0.002066115703) and add it to both sides. Add '0.002066115703' to each side of the equation. -0.09090909091z + 0.002066115703 + z2 = 1.636363636 + 0.002066115703 Reorder the terms: 0.002066115703 + -0.09090909091z + z2 = 1.636363636 + 0.002066115703 Combine like terms: 1.636363636 + 0.002066115703 = 1.638429751703 0.002066115703 + -0.09090909091z + z2 = 1.638429751703 Factor a perfect square on the left side: (z + -0.04545454546)(z + -0.04545454546) = 1.638429751703 Calculate the square root of the right side: 1.280011622 Break this problem into two subproblems by setting (z + -0.04545454546) equal to 1.280011622 and -1.280011622.Subproblem 1
z + -0.04545454546 = 1.280011622 Simplifying z + -0.04545454546 = 1.280011622 Reorder the terms: -0.04545454546 + z = 1.280011622 Solving -0.04545454546 + z = 1.280011622 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.04545454546' to each side of the equation. -0.04545454546 + 0.04545454546 + z = 1.280011622 + 0.04545454546 Combine like terms: -0.04545454546 + 0.04545454546 = 0.00000000000 0.00000000000 + z = 1.280011622 + 0.04545454546 z = 1.280011622 + 0.04545454546 Combine like terms: 1.280011622 + 0.04545454546 = 1.32546616746 z = 1.32546616746 Simplifying z = 1.32546616746Subproblem 2
z + -0.04545454546 = -1.280011622 Simplifying z + -0.04545454546 = -1.280011622 Reorder the terms: -0.04545454546 + z = -1.280011622 Solving -0.04545454546 + z = -1.280011622 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '0.04545454546' to each side of the equation. -0.04545454546 + 0.04545454546 + z = -1.280011622 + 0.04545454546 Combine like terms: -0.04545454546 + 0.04545454546 = 0.00000000000 0.00000000000 + z = -1.280011622 + 0.04545454546 z = -1.280011622 + 0.04545454546 Combine like terms: -1.280011622 + 0.04545454546 = -1.23455707654 z = -1.23455707654 Simplifying z = -1.23455707654Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.32546616746, -1.23455707654}Solution
z = {1.32546616746, -1.23455707654}
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