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Simplifying 3a2 + a = 7 Reorder the terms: a + 3a2 = 7 Solving a + 3a2 = 7 Solving for variable 'a'. Reorder the terms: -7 + a + 3a2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.333333333 + 0.3333333333a + a2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + 0.3333333333a + 2.333333333 + a2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + 0.3333333333a + a2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + 0.3333333333a + a2 = 0 + 2.333333333 0.3333333333a + a2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 0.3333333333a + a2 = 2.333333333 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.3333333333a + 0.25 + a2 = 2.333333333 + 0.25 Reorder the terms: 0.25 + 0.3333333333a + a2 = 2.333333333 + 0.25 Combine like terms: 2.333333333 + 0.25 = 2.583333333 0.25 + 0.3333333333a + a2 = 2.583333333 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 2.583333333 Calculate the square root of the right side: 1.607275127 Break this problem into two subproblems by setting (a + 0.5) equal to 1.607275127 and -1.607275127.Subproblem 1
a + 0.5 = 1.607275127 Simplifying a + 0.5 = 1.607275127 Reorder the terms: 0.5 + a = 1.607275127 Solving 0.5 + a = 1.607275127 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 1.607275127 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.607275127 + -0.5 a = 1.607275127 + -0.5 Combine like terms: 1.607275127 + -0.5 = 1.107275127 a = 1.107275127 Simplifying a = 1.107275127Subproblem 2
a + 0.5 = -1.607275127 Simplifying a + 0.5 = -1.607275127 Reorder the terms: 0.5 + a = -1.607275127 Solving 0.5 + a = -1.607275127 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -1.607275127 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.607275127 + -0.5 a = -1.607275127 + -0.5 Combine like terms: -1.607275127 + -0.5 = -2.107275127 a = -2.107275127 Simplifying a = -2.107275127Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.107275127, -2.107275127}
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