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Simplifying 24a2 + -61a + -3 = 0 Reorder the terms: -3 + -61a + 24a2 = 0 Solving -3 + -61a + 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 + -2.541666667a + a2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -2.541666667a + 0.125 + a2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -2.541666667a + a2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -2.541666667a + a2 = 0 + 0.125 -2.541666667a + a2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -2.541666667a + a2 = 0.125 The a term is -2.541666667a. Take half its coefficient (-1.270833334). Square it (1.615017363) and add it to both sides. Add '1.615017363' to each side of the equation. -2.541666667a + 1.615017363 + a2 = 0.125 + 1.615017363 Reorder the terms: 1.615017363 + -2.541666667a + a2 = 0.125 + 1.615017363 Combine like terms: 0.125 + 1.615017363 = 1.740017363 1.615017363 + -2.541666667a + a2 = 1.740017363 Factor a perfect square on the left side: (a + -1.270833334)(a + -1.270833334) = 1.740017363 Calculate the square root of the right side: 1.319097177 Break this problem into two subproblems by setting (a + -1.270833334) equal to 1.319097177 and -1.319097177.Subproblem 1
a + -1.270833334 = 1.319097177 Simplifying a + -1.270833334 = 1.319097177 Reorder the terms: -1.270833334 + a = 1.319097177 Solving -1.270833334 + a = 1.319097177 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.270833334' to each side of the equation. -1.270833334 + 1.270833334 + a = 1.319097177 + 1.270833334 Combine like terms: -1.270833334 + 1.270833334 = 0.000000000 0.000000000 + a = 1.319097177 + 1.270833334 a = 1.319097177 + 1.270833334 Combine like terms: 1.319097177 + 1.270833334 = 2.589930511 a = 2.589930511 Simplifying a = 2.589930511Subproblem 2
a + -1.270833334 = -1.319097177 Simplifying a + -1.270833334 = -1.319097177 Reorder the terms: -1.270833334 + a = -1.319097177 Solving -1.270833334 + a = -1.319097177 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.270833334' to each side of the equation. -1.270833334 + 1.270833334 + a = -1.319097177 + 1.270833334 Combine like terms: -1.270833334 + 1.270833334 = 0.000000000 0.000000000 + a = -1.319097177 + 1.270833334 a = -1.319097177 + 1.270833334 Combine like terms: -1.319097177 + 1.270833334 = -0.048263843 a = -0.048263843 Simplifying a = -0.048263843Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.589930511, -0.048263843}
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