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Simplifying (0.4a + -0.1)(0.07a + 1) = 0 Reorder the terms: (-0.1 + 0.4a)(0.07a + 1) = 0 Reorder the terms: (-0.1 + 0.4a)(1 + 0.07a) = 0 Multiply (-0.1 + 0.4a) * (1 + 0.07a) (-0.1(1 + 0.07a) + 0.4a * (1 + 0.07a)) = 0 ((1 * -0.1 + 0.07a * -0.1) + 0.4a * (1 + 0.07a)) = 0 ((-0.1 + -0.007a) + 0.4a * (1 + 0.07a)) = 0 (-0.1 + -0.007a + (1 * 0.4a + 0.07a * 0.4a)) = 0 (-0.1 + -0.007a + (0.4a + 0.028a2)) = 0 Combine like terms: -0.007a + 0.4a = 0.393a (-0.1 + 0.393a + 0.028a2) = 0 Solving -0.1 + 0.393a + 0.028a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.028 the coefficient of the squared term: Divide each side by '0.028'. -3.571428571 + 14.03571429a + a2 = 0 Move the constant term to the right: Add '3.571428571' to each side of the equation. -3.571428571 + 14.03571429a + 3.571428571 + a2 = 0 + 3.571428571 Reorder the terms: -3.571428571 + 3.571428571 + 14.03571429a + a2 = 0 + 3.571428571 Combine like terms: -3.571428571 + 3.571428571 = 0.000000000 0.000000000 + 14.03571429a + a2 = 0 + 3.571428571 14.03571429a + a2 = 0 + 3.571428571 Combine like terms: 0 + 3.571428571 = 3.571428571 14.03571429a + a2 = 3.571428571 The a term is 14.03571429a. Take half its coefficient (7.017857145). Square it (49.25031891) and add it to both sides. Add '49.25031891' to each side of the equation. 14.03571429a + 49.25031891 + a2 = 3.571428571 + 49.25031891 Reorder the terms: 49.25031891 + 14.03571429a + a2 = 3.571428571 + 49.25031891 Combine like terms: 3.571428571 + 49.25031891 = 52.821747481 49.25031891 + 14.03571429a + a2 = 52.821747481 Factor a perfect square on the left side: (a + 7.017857145)(a + 7.017857145) = 52.821747481 Calculate the square root of the right side: 7.267857145 Break this problem into two subproblems by setting (a + 7.017857145) equal to 7.267857145 and -7.267857145.Subproblem 1
a + 7.017857145 = 7.267857145 Simplifying a + 7.017857145 = 7.267857145 Reorder the terms: 7.017857145 + a = 7.267857145 Solving 7.017857145 + a = 7.267857145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-7.017857145' to each side of the equation. 7.017857145 + -7.017857145 + a = 7.267857145 + -7.017857145 Combine like terms: 7.017857145 + -7.017857145 = 0.000000000 0.000000000 + a = 7.267857145 + -7.017857145 a = 7.267857145 + -7.017857145 Combine like terms: 7.267857145 + -7.017857145 = 0.25 a = 0.25 Simplifying a = 0.25Subproblem 2
a + 7.017857145 = -7.267857145 Simplifying a + 7.017857145 = -7.267857145 Reorder the terms: 7.017857145 + a = -7.267857145 Solving 7.017857145 + a = -7.267857145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-7.017857145' to each side of the equation. 7.017857145 + -7.017857145 + a = -7.267857145 + -7.017857145 Combine like terms: 7.017857145 + -7.017857145 = 0.000000000 0.000000000 + a = -7.267857145 + -7.017857145 a = -7.267857145 + -7.017857145 Combine like terms: -7.267857145 + -7.017857145 = -14.28571429 a = -14.28571429 Simplifying a = -14.28571429Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.25, -14.28571429}
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