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Simplifying 4a2 + 11a + 3 = 0 Reorder the terms: 3 + 11a + 4a2 = 0 Solving 3 + 11a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.75 + 2.75a + a2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + 2.75a + -0.75 + a2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + 2.75a + a2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + 2.75a + a2 = 0 + -0.75 2.75a + a2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 2.75a + a2 = -0.75 The a term is 2.75a. Take half its coefficient (1.375). Square it (1.890625) and add it to both sides. Add '1.890625' to each side of the equation. 2.75a + 1.890625 + a2 = -0.75 + 1.890625 Reorder the terms: 1.890625 + 2.75a + a2 = -0.75 + 1.890625 Combine like terms: -0.75 + 1.890625 = 1.140625 1.890625 + 2.75a + a2 = 1.140625 Factor a perfect square on the left side: (a + 1.375)(a + 1.375) = 1.140625 Calculate the square root of the right side: 1.068000468 Break this problem into two subproblems by setting (a + 1.375) equal to 1.068000468 and -1.068000468.Subproblem 1
a + 1.375 = 1.068000468 Simplifying a + 1.375 = 1.068000468 Reorder the terms: 1.375 + a = 1.068000468 Solving 1.375 + a = 1.068000468 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + a = 1.068000468 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + a = 1.068000468 + -1.375 a = 1.068000468 + -1.375 Combine like terms: 1.068000468 + -1.375 = -0.306999532 a = -0.306999532 Simplifying a = -0.306999532Subproblem 2
a + 1.375 = -1.068000468 Simplifying a + 1.375 = -1.068000468 Reorder the terms: 1.375 + a = -1.068000468 Solving 1.375 + a = -1.068000468 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.375' to each side of the equation. 1.375 + -1.375 + a = -1.068000468 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + a = -1.068000468 + -1.375 a = -1.068000468 + -1.375 Combine like terms: -1.068000468 + -1.375 = -2.443000468 a = -2.443000468 Simplifying a = -2.443000468Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.306999532, -2.443000468}
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