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Simplifying 3a2 = 11a + 3 Reorder the terms: 3a2 = 3 + 11a Solving 3a2 = 3 + 11a Solving for variable 'a'. Reorder the terms: -3 + -11a + 3a2 = 3 + 11a + -3 + -11a Reorder the terms: -3 + -11a + 3a2 = 3 + -3 + 11a + -11a Combine like terms: 3 + -3 = 0 -3 + -11a + 3a2 = 0 + 11a + -11a -3 + -11a + 3a2 = 11a + -11a Combine like terms: 11a + -11a = 0 -3 + -11a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1 + -3.666666667a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -3.666666667a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -3.666666667a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -3.666666667a + a2 = 0 + 1 -3.666666667a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -3.666666667a + a2 = 1 The a term is -3.666666667a. Take half its coefficient (-1.833333334). Square it (3.361111114) and add it to both sides. Add '3.361111114' to each side of the equation. -3.666666667a + 3.361111114 + a2 = 1 + 3.361111114 Reorder the terms: 3.361111114 + -3.666666667a + a2 = 1 + 3.361111114 Combine like terms: 1 + 3.361111114 = 4.361111114 3.361111114 + -3.666666667a + a2 = 4.361111114 Factor a perfect square on the left side: (a + -1.833333334)(a + -1.833333334) = 4.361111114 Calculate the square root of the right side: 2.088327348 Break this problem into two subproblems by setting (a + -1.833333334) equal to 2.088327348 and -2.088327348.Subproblem 1
a + -1.833333334 = 2.088327348 Simplifying a + -1.833333334 = 2.088327348 Reorder the terms: -1.833333334 + a = 2.088327348 Solving -1.833333334 + a = 2.088327348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = 2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = 2.088327348 + 1.833333334 a = 2.088327348 + 1.833333334 Combine like terms: 2.088327348 + 1.833333334 = 3.921660682 a = 3.921660682 Simplifying a = 3.921660682Subproblem 2
a + -1.833333334 = -2.088327348 Simplifying a + -1.833333334 = -2.088327348 Reorder the terms: -1.833333334 + a = -2.088327348 Solving -1.833333334 + a = -2.088327348 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.833333334' to each side of the equation. -1.833333334 + 1.833333334 + a = -2.088327348 + 1.833333334 Combine like terms: -1.833333334 + 1.833333334 = 0.000000000 0.000000000 + a = -2.088327348 + 1.833333334 a = -2.088327348 + 1.833333334 Combine like terms: -2.088327348 + 1.833333334 = -0.254994014 a = -0.254994014 Simplifying a = -0.254994014Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.921660682, -0.254994014}
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