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Simplifying 4a2 + 11a + 5 = 0 Reorder the terms: 5 + 11a + 4a2 = 0 Solving 5 + 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'. 1.25 + 2.75a + a2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + 2.75a + -1.25 + a2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + 2.75a + a2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + 2.75a + a2 = 0 + -1.25 2.75a + a2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 2.75a + a2 = -1.25 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 = -1.25 + 1.890625 Reorder the terms: 1.890625 + 2.75a + a2 = -1.25 + 1.890625 Combine like terms: -1.25 + 1.890625 = 0.640625 1.890625 + 2.75a + a2 = 0.640625 Factor a perfect square on the left side: (a + 1.375)(a + 1.375) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (a + 1.375) equal to 0.80039053 and -0.80039053.Subproblem 1
a + 1.375 = 0.80039053 Simplifying a + 1.375 = 0.80039053 Reorder the terms: 1.375 + a = 0.80039053 Solving 1.375 + a = 0.80039053 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 = 0.80039053 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + a = 0.80039053 + -1.375 a = 0.80039053 + -1.375 Combine like terms: 0.80039053 + -1.375 = -0.57460947 a = -0.57460947 Simplifying a = -0.57460947Subproblem 2
a + 1.375 = -0.80039053 Simplifying a + 1.375 = -0.80039053 Reorder the terms: 1.375 + a = -0.80039053 Solving 1.375 + a = -0.80039053 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 = -0.80039053 + -1.375 Combine like terms: 1.375 + -1.375 = 0.000 0.000 + a = -0.80039053 + -1.375 a = -0.80039053 + -1.375 Combine like terms: -0.80039053 + -1.375 = -2.17539053 a = -2.17539053 Simplifying a = -2.17539053Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.57460947, -2.17539053}
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