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Simplifying 4a(80 + -2a) = 0.24 (80 * 4a + -2a * 4a) = 0.24 (320a + -8a2) = 0.24 Solving 320a + -8a2 = 0.24 Solving for variable 'a'. Reorder the terms: -0.24 + 320a + -8a2 = 0.24 + -0.24 Combine like terms: 0.24 + -0.24 = 0.00 -0.24 + 320a + -8a2 = 0.00 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. 0.03 + -40a + a2 = 0.00 Move the constant term to the right: Add '-0.03' to each side of the equation. 0.03 + -40a + -0.03 + a2 = 0.00 + -0.03 Reorder the terms: 0.03 + -0.03 + -40a + a2 = 0.00 + -0.03 Combine like terms: 0.03 + -0.03 = 0.00 0.00 + -40a + a2 = 0.00 + -0.03 -40a + a2 = 0.00 + -0.03 Combine like terms: 0.00 + -0.03 = -0.03 -40a + a2 = -0.03 The a term is -40a. Take half its coefficient (-20). Square it (400) and add it to both sides. Add '400' to each side of the equation. -40a + 400 + a2 = -0.03 + 400 Reorder the terms: 400 + -40a + a2 = -0.03 + 400 Combine like terms: -0.03 + 400 = 399.97 400 + -40a + a2 = 399.97 Factor a perfect square on the left side: (a + -20)(a + -20) = 399.97 Calculate the square root of the right side: 19.999249986 Break this problem into two subproblems by setting (a + -20) equal to 19.999249986 and -19.999249986.Subproblem 1
a + -20 = 19.999249986 Simplifying a + -20 = 19.999249986 Reorder the terms: -20 + a = 19.999249986 Solving -20 + a = 19.999249986 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + a = 19.999249986 + 20 Combine like terms: -20 + 20 = 0 0 + a = 19.999249986 + 20 a = 19.999249986 + 20 Combine like terms: 19.999249986 + 20 = 39.999249986 a = 39.999249986 Simplifying a = 39.999249986Subproblem 2
a + -20 = -19.999249986 Simplifying a + -20 = -19.999249986 Reorder the terms: -20 + a = -19.999249986 Solving -20 + a = -19.999249986 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '20' to each side of the equation. -20 + 20 + a = -19.999249986 + 20 Combine like terms: -20 + 20 = 0 0 + a = -19.999249986 + 20 a = -19.999249986 + 20 Combine like terms: -19.999249986 + 20 = 0.000750014 a = 0.000750014 Simplifying a = 0.000750014Solution
The solution to the problem is based on the solutions from the subproblems. a = {39.999249986, 0.000750014}
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