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Simplifying 4.571428571a2 = 1 + -7.428571429a Solving 4.571428571a2 = 1 + -7.428571429a Solving for variable 'a'. Reorder the terms: -1 + 7.428571429a + 4.571428571a2 = 1 + -7.428571429a + -1 + 7.428571429a Reorder the terms: -1 + 7.428571429a + 4.571428571a2 = 1 + -1 + -7.428571429a + 7.428571429a Combine like terms: 1 + -1 = 0 -1 + 7.428571429a + 4.571428571a2 = 0 + -7.428571429a + 7.428571429a -1 + 7.428571429a + 4.571428571a2 = -7.428571429a + 7.428571429a Combine like terms: -7.428571429a + 7.428571429a = 0.000000000 -1 + 7.428571429a + 4.571428571a2 = 0.000000000 Begin completing the square. Divide all terms by 4.571428571 the coefficient of the squared term: Divide each side by '4.571428571'. -0.21875 + 1.625a + a2 = 0 Move the constant term to the right: Add '0.21875' to each side of the equation. -0.21875 + 1.625a + 0.21875 + a2 = 0 + 0.21875 Reorder the terms: -0.21875 + 0.21875 + 1.625a + a2 = 0 + 0.21875 Combine like terms: -0.21875 + 0.21875 = 0.00000 0.00000 + 1.625a + a2 = 0 + 0.21875 1.625a + a2 = 0 + 0.21875 Combine like terms: 0 + 0.21875 = 0.21875 1.625a + a2 = 0.21875 The a term is 1.625a. Take half its coefficient (0.8125). Square it (0.66015625) and add it to both sides. Add '0.66015625' to each side of the equation. 1.625a + 0.66015625 + a2 = 0.21875 + 0.66015625 Reorder the terms: 0.66015625 + 1.625a + a2 = 0.21875 + 0.66015625 Combine like terms: 0.21875 + 0.66015625 = 0.87890625 0.66015625 + 1.625a + a2 = 0.87890625 Factor a perfect square on the left side: (a + 0.8125)(a + 0.8125) = 0.87890625 Calculate the square root of the right side: 0.9375 Break this problem into two subproblems by setting (a + 0.8125) equal to 0.9375 and -0.9375.Subproblem 1
a + 0.8125 = 0.9375 Simplifying a + 0.8125 = 0.9375 Reorder the terms: 0.8125 + a = 0.9375 Solving 0.8125 + a = 0.9375 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + a = 0.9375 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + a = 0.9375 + -0.8125 a = 0.9375 + -0.8125 Combine like terms: 0.9375 + -0.8125 = 0.125 a = 0.125 Simplifying a = 0.125Subproblem 2
a + 0.8125 = -0.9375 Simplifying a + 0.8125 = -0.9375 Reorder the terms: 0.8125 + a = -0.9375 Solving 0.8125 + a = -0.9375 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + a = -0.9375 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + a = -0.9375 + -0.8125 a = -0.9375 + -0.8125 Combine like terms: -0.9375 + -0.8125 = -1.75 a = -1.75 Simplifying a = -1.75Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.125, -1.75}
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