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Simplifying 20 + -1a = 2a(6a + 1) Reorder the terms: 20 + -1a = 2a(1 + 6a) 20 + -1a = (1 * 2a + 6a * 2a) 20 + -1a = (2a + 12a2) Solving 20 + -1a = 2a + 12a2 Solving for variable 'a'. Combine like terms: -1a + -2a = -3a 20 + -3a + -12a2 = 2a + 12a2 + -2a + -12a2 Reorder the terms: 20 + -3a + -12a2 = 2a + -2a + 12a2 + -12a2 Combine like terms: 2a + -2a = 0 20 + -3a + -12a2 = 0 + 12a2 + -12a2 20 + -3a + -12a2 = 12a2 + -12a2 Combine like terms: 12a2 + -12a2 = 0 20 + -3a + -12a2 = 0 Begin completing the square. Divide all terms by -12 the coefficient of the squared term: Divide each side by '-12'. -1.666666667 + 0.25a + a2 = 0 Move the constant term to the right: Add '1.666666667' to each side of the equation. -1.666666667 + 0.25a + 1.666666667 + a2 = 0 + 1.666666667 Reorder the terms: -1.666666667 + 1.666666667 + 0.25a + a2 = 0 + 1.666666667 Combine like terms: -1.666666667 + 1.666666667 = 0.000000000 0.000000000 + 0.25a + a2 = 0 + 1.666666667 0.25a + a2 = 0 + 1.666666667 Combine like terms: 0 + 1.666666667 = 1.666666667 0.25a + a2 = 1.666666667 The a term is 0.25a. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25a + 0.015625 + a2 = 1.666666667 + 0.015625 Reorder the terms: 0.015625 + 0.25a + a2 = 1.666666667 + 0.015625 Combine like terms: 1.666666667 + 0.015625 = 1.682291667 0.015625 + 0.25a + a2 = 1.682291667 Factor a perfect square on the left side: (a + 0.125)(a + 0.125) = 1.682291667 Calculate the square root of the right side: 1.297031868 Break this problem into two subproblems by setting (a + 0.125) equal to 1.297031868 and -1.297031868.Subproblem 1
a + 0.125 = 1.297031868 Simplifying a + 0.125 = 1.297031868 Reorder the terms: 0.125 + a = 1.297031868 Solving 0.125 + a = 1.297031868 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = 1.297031868 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = 1.297031868 + -0.125 a = 1.297031868 + -0.125 Combine like terms: 1.297031868 + -0.125 = 1.172031868 a = 1.172031868 Simplifying a = 1.172031868Subproblem 2
a + 0.125 = -1.297031868 Simplifying a + 0.125 = -1.297031868 Reorder the terms: 0.125 + a = -1.297031868 Solving 0.125 + a = -1.297031868 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + a = -1.297031868 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + a = -1.297031868 + -0.125 a = -1.297031868 + -0.125 Combine like terms: -1.297031868 + -0.125 = -1.422031868 a = -1.422031868 Simplifying a = -1.422031868Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.172031868, -1.422031868}
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