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Simplifying (0.2 + -3a)(0.7 + -8a) = 0 Multiply (0.2 + -3a) * (0.7 + -8a) (0.2(0.7 + -8a) + -3a * (0.7 + -8a)) = 0 ((0.7 * 0.2 + -8a * 0.2) + -3a * (0.7 + -8a)) = 0 ((0.14 + -1.6a) + -3a * (0.7 + -8a)) = 0 (0.14 + -1.6a + (0.7 * -3a + -8a * -3a)) = 0 (0.14 + -1.6a + (-2.1a + 24a2)) = 0 Combine like terms: -1.6a + -2.1a = -3.7a (0.14 + -3.7a + 24a2) = 0 Solving 0.14 + -3.7a + 24a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 0.005833333333 + -0.1541666667a + a2 = 0 Move the constant term to the right: Add '-0.005833333333' to each side of the equation. 0.005833333333 + -0.1541666667a + -0.005833333333 + a2 = 0 + -0.005833333333 Reorder the terms: 0.005833333333 + -0.005833333333 + -0.1541666667a + a2 = 0 + -0.005833333333 Combine like terms: 0.005833333333 + -0.005833333333 = 0.000000000000 0.000000000000 + -0.1541666667a + a2 = 0 + -0.005833333333 -0.1541666667a + a2 = 0 + -0.005833333333 Combine like terms: 0 + -0.005833333333 = -0.005833333333 -0.1541666667a + a2 = -0.005833333333 The a term is -0.1541666667a. Take half its coefficient (-0.07708333335). Square it (0.005941840280) and add it to both sides. Add '0.005941840280' to each side of the equation. -0.1541666667a + 0.005941840280 + a2 = -0.005833333333 + 0.005941840280 Reorder the terms: 0.005941840280 + -0.1541666667a + a2 = -0.005833333333 + 0.005941840280 Combine like terms: -0.005833333333 + 0.005941840280 = 0.000108506947 0.005941840280 + -0.1541666667a + a2 = 0.000108506947 Factor a perfect square on the left side: (a + -0.07708333335)(a + -0.07708333335) = 0.000108506947 Calculate the square root of the right side: 0.010416667 Break this problem into two subproblems by setting (a + -0.07708333335) equal to 0.010416667 and -0.010416667.Subproblem 1
a + -0.07708333335 = 0.010416667 Simplifying a + -0.07708333335 = 0.010416667 Reorder the terms: -0.07708333335 + a = 0.010416667 Solving -0.07708333335 + a = 0.010416667 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.07708333335' to each side of the equation. -0.07708333335 + 0.07708333335 + a = 0.010416667 + 0.07708333335 Combine like terms: -0.07708333335 + 0.07708333335 = 0.00000000000 0.00000000000 + a = 0.010416667 + 0.07708333335 a = 0.010416667 + 0.07708333335 Combine like terms: 0.010416667 + 0.07708333335 = 0.08750000035 a = 0.08750000035 Simplifying a = 0.08750000035Subproblem 2
a + -0.07708333335 = -0.010416667 Simplifying a + -0.07708333335 = -0.010416667 Reorder the terms: -0.07708333335 + a = -0.010416667 Solving -0.07708333335 + a = -0.010416667 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.07708333335' to each side of the equation. -0.07708333335 + 0.07708333335 + a = -0.010416667 + 0.07708333335 Combine like terms: -0.07708333335 + 0.07708333335 = 0.00000000000 0.00000000000 + a = -0.010416667 + 0.07708333335 a = -0.010416667 + 0.07708333335 Combine like terms: -0.010416667 + 0.07708333335 = 0.06666666635 a = 0.06666666635 Simplifying a = 0.06666666635Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.08750000035, 0.06666666635}
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