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Simplifying (4a + 3) * a = 6 Reorder the terms: (3 + 4a) * a = 6 Reorder the terms for easier multiplication: a(3 + 4a) = 6 (3 * a + 4a * a) = 6 (3a + 4a2) = 6 Solving 3a + 4a2 = 6 Solving for variable 'a'. Reorder the terms: -6 + 3a + 4a2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 3a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.5 + 0.75a + a2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 0.75a + 1.5 + a2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 0.75a + a2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 0.75a + a2 = 0 + 1.5 0.75a + a2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 0.75a + a2 = 1.5 The a term is 0.75a. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75a + 0.140625 + a2 = 1.5 + 0.140625 Reorder the terms: 0.140625 + 0.75a + a2 = 1.5 + 0.140625 Combine like terms: 1.5 + 0.140625 = 1.640625 0.140625 + 0.75a + a2 = 1.640625 Factor a perfect square on the left side: (a + 0.375)(a + 0.375) = 1.640625 Calculate the square root of the right side: 1.280868846 Break this problem into two subproblems by setting (a + 0.375) equal to 1.280868846 and -1.280868846.Subproblem 1
a + 0.375 = 1.280868846 Simplifying a + 0.375 = 1.280868846 Reorder the terms: 0.375 + a = 1.280868846 Solving 0.375 + a = 1.280868846 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = 1.280868846 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = 1.280868846 + -0.375 a = 1.280868846 + -0.375 Combine like terms: 1.280868846 + -0.375 = 0.905868846 a = 0.905868846 Simplifying a = 0.905868846Subproblem 2
a + 0.375 = -1.280868846 Simplifying a + 0.375 = -1.280868846 Reorder the terms: 0.375 + a = -1.280868846 Solving 0.375 + a = -1.280868846 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + a = -1.280868846 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + a = -1.280868846 + -0.375 a = -1.280868846 + -0.375 Combine like terms: -1.280868846 + -0.375 = -1.655868846 a = -1.655868846 Simplifying a = -1.655868846Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.905868846, -1.655868846}
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