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Simplifying 4a2 + 28a2 + a + 7 = 0 Reorder the terms: 7 + a + 4a2 + 28a2 = 0 Combine like terms: 4a2 + 28a2 = 32a2 7 + a + 32a2 = 0 Solving 7 + a + 32a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. 0.21875 + 0.03125a + a2 = 0 Move the constant term to the right: Add '-0.21875' to each side of the equation. 0.21875 + 0.03125a + -0.21875 + a2 = 0 + -0.21875 Reorder the terms: 0.21875 + -0.21875 + 0.03125a + a2 = 0 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + 0.03125a + a2 = 0 + -0.21875 0.03125a + a2 = 0 + -0.21875 Combine like terms: 0 + -0.21875 = -0.21875 0.03125a + a2 = -0.21875 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. 0.03125a + 0.25 + a2 = -0.21875 + 0.25 Reorder the terms: 0.25 + 0.03125a + a2 = -0.21875 + 0.25 Combine like terms: -0.21875 + 0.25 = 0.03125 0.25 + 0.03125a + a2 = 0.03125 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 0.03125 Calculate the square root of the right side: 0.176776695 Break this problem into two subproblems by setting (a + 0.5) equal to 0.176776695 and -0.176776695.Subproblem 1
a + 0.5 = 0.176776695 Simplifying a + 0.5 = 0.176776695 Reorder the terms: 0.5 + a = 0.176776695 Solving 0.5 + a = 0.176776695 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 = 0.176776695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 0.176776695 + -0.5 a = 0.176776695 + -0.5 Combine like terms: 0.176776695 + -0.5 = -0.323223305 a = -0.323223305 Simplifying a = -0.323223305Subproblem 2
a + 0.5 = -0.176776695 Simplifying a + 0.5 = -0.176776695 Reorder the terms: 0.5 + a = -0.176776695 Solving 0.5 + a = -0.176776695 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 = -0.176776695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -0.176776695 + -0.5 a = -0.176776695 + -0.5 Combine like terms: -0.176776695 + -0.5 = -0.676776695 a = -0.676776695 Simplifying a = -0.676776695Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.323223305, -0.676776695}
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