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Simplifying 24z2 + 55z + 24 = 0 Reorder the terms: 24 + 55z + 24z2 = 0 Solving 24 + 55z + 24z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 1 + 2.291666667z + z2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.291666667z + -1 + z2 = 0 + -1 Reorder the terms: 1 + -1 + 2.291666667z + z2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.291666667z + z2 = 0 + -1 2.291666667z + z2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.291666667z + z2 = -1 The z term is 2.291666667z. Take half its coefficient (1.145833334). Square it (1.312934029) and add it to both sides. Add '1.312934029' to each side of the equation. 2.291666667z + 1.312934029 + z2 = -1 + 1.312934029 Reorder the terms: 1.312934029 + 2.291666667z + z2 = -1 + 1.312934029 Combine like terms: -1 + 1.312934029 = 0.312934029 1.312934029 + 2.291666667z + z2 = 0.312934029 Factor a perfect square on the left side: (z + 1.145833334)(z + 1.145833334) = 0.312934029 Calculate the square root of the right side: 0.559405067 Break this problem into two subproblems by setting (z + 1.145833334) equal to 0.559405067 and -0.559405067.Subproblem 1
z + 1.145833334 = 0.559405067 Simplifying z + 1.145833334 = 0.559405067 Reorder the terms: 1.145833334 + z = 0.559405067 Solving 1.145833334 + z = 0.559405067 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.145833334' to each side of the equation. 1.145833334 + -1.145833334 + z = 0.559405067 + -1.145833334 Combine like terms: 1.145833334 + -1.145833334 = 0.000000000 0.000000000 + z = 0.559405067 + -1.145833334 z = 0.559405067 + -1.145833334 Combine like terms: 0.559405067 + -1.145833334 = -0.586428267 z = -0.586428267 Simplifying z = -0.586428267Subproblem 2
z + 1.145833334 = -0.559405067 Simplifying z + 1.145833334 = -0.559405067 Reorder the terms: 1.145833334 + z = -0.559405067 Solving 1.145833334 + z = -0.559405067 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1.145833334' to each side of the equation. 1.145833334 + -1.145833334 + z = -0.559405067 + -1.145833334 Combine like terms: 1.145833334 + -1.145833334 = 0.000000000 0.000000000 + z = -0.559405067 + -1.145833334 z = -0.559405067 + -1.145833334 Combine like terms: -0.559405067 + -1.145833334 = -1.705238401 z = -1.705238401 Simplifying z = -1.705238401Solution
The solution to the problem is based on the solutions from the subproblems. z = {-0.586428267, -1.705238401}
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