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Simplifying 24a2 + -55a + 3 = 0 Reorder the terms: 3 + -55a + 24a2 = 0 Solving 3 + -55a + 24a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.125 + -2.291666667a + a2 = 0 Move the constant term to the right: Add '-0.125' to each side of the equation. 0.125 + -2.291666667a + -0.125 + a2 = 0 + -0.125 Reorder the terms: 0.125 + -0.125 + -2.291666667a + a2 = 0 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + -2.291666667a + a2 = 0 + -0.125 -2.291666667a + a2 = 0 + -0.125 Combine like terms: 0 + -0.125 = -0.125 -2.291666667a + a2 = -0.125 The a term is -2.291666667a. 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. -2.291666667a + 1.312934029 + a2 = -0.125 + 1.312934029 Reorder the terms: 1.312934029 + -2.291666667a + a2 = -0.125 + 1.312934029 Combine like terms: -0.125 + 1.312934029 = 1.187934029 1.312934029 + -2.291666667a + a2 = 1.187934029 Factor a perfect square on the left side: (a + -1.145833334)(a + -1.145833334) = 1.187934029 Calculate the square root of the right side: 1.089923864 Break this problem into two subproblems by setting (a + -1.145833334) equal to 1.089923864 and -1.089923864.Subproblem 1
a + -1.145833334 = 1.089923864 Simplifying a + -1.145833334 = 1.089923864 Reorder the terms: -1.145833334 + a = 1.089923864 Solving -1.145833334 + a = 1.089923864 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.145833334' to each side of the equation. -1.145833334 + 1.145833334 + a = 1.089923864 + 1.145833334 Combine like terms: -1.145833334 + 1.145833334 = 0.000000000 0.000000000 + a = 1.089923864 + 1.145833334 a = 1.089923864 + 1.145833334 Combine like terms: 1.089923864 + 1.145833334 = 2.235757198 a = 2.235757198 Simplifying a = 2.235757198Subproblem 2
a + -1.145833334 = -1.089923864 Simplifying a + -1.145833334 = -1.089923864 Reorder the terms: -1.145833334 + a = -1.089923864 Solving -1.145833334 + a = -1.089923864 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.145833334' to each side of the equation. -1.145833334 + 1.145833334 + a = -1.089923864 + 1.145833334 Combine like terms: -1.145833334 + 1.145833334 = 0.000000000 0.000000000 + a = -1.089923864 + 1.145833334 a = -1.089923864 + 1.145833334 Combine like terms: -1.089923864 + 1.145833334 = 0.05590947 a = 0.05590947 Simplifying a = 0.05590947Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.235757198, 0.05590947}
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