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Simplifying 0.14(a + -3)(7a + 4) = 0 Reorder the terms: 0.14(-3 + a)(7a + 4) = 0 Reorder the terms: 0.14(-3 + a)(4 + 7a) = 0 Multiply (-3 + a) * (4 + 7a) 0.14(-3(4 + 7a) + a(4 + 7a)) = 0 0.14((4 * -3 + 7a * -3) + a(4 + 7a)) = 0 0.14((-12 + -21a) + a(4 + 7a)) = 0 0.14(-12 + -21a + (4 * a + 7a * a)) = 0 0.14(-12 + -21a + (4a + 7a2)) = 0 Combine like terms: -21a + 4a = -17a 0.14(-12 + -17a + 7a2) = 0 (-12 * 0.14 + -17a * 0.14 + 7a2 * 0.14) = 0 (-1.68 + -2.38a + 0.98a2) = 0 Solving -1.68 + -2.38a + 0.98a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.98 the coefficient of the squared term: Divide each side by '0.98'. -1.714285714 + -2.428571429a + a2 = 0 Move the constant term to the right: Add '1.714285714' to each side of the equation. -1.714285714 + -2.428571429a + 1.714285714 + a2 = 0 + 1.714285714 Reorder the terms: -1.714285714 + 1.714285714 + -2.428571429a + a2 = 0 + 1.714285714 Combine like terms: -1.714285714 + 1.714285714 = 0.000000000 0.000000000 + -2.428571429a + a2 = 0 + 1.714285714 -2.428571429a + a2 = 0 + 1.714285714 Combine like terms: 0 + 1.714285714 = 1.714285714 -2.428571429a + a2 = 1.714285714 The a term is -2.428571429a. Take half its coefficient (-1.214285715). Square it (1.474489798) and add it to both sides. Add '1.474489798' to each side of the equation. -2.428571429a + 1.474489798 + a2 = 1.714285714 + 1.474489798 Reorder the terms: 1.474489798 + -2.428571429a + a2 = 1.714285714 + 1.474489798 Combine like terms: 1.714285714 + 1.474489798 = 3.188775512 1.474489798 + -2.428571429a + a2 = 3.188775512 Factor a perfect square on the left side: (a + -1.214285715)(a + -1.214285715) = 3.188775512 Calculate the square root of the right side: 1.785714286 Break this problem into two subproblems by setting (a + -1.214285715) equal to 1.785714286 and -1.785714286.Subproblem 1
a + -1.214285715 = 1.785714286 Simplifying a + -1.214285715 = 1.785714286 Reorder the terms: -1.214285715 + a = 1.785714286 Solving -1.214285715 + a = 1.785714286 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + a = 1.785714286 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + a = 1.785714286 + 1.214285715 a = 1.785714286 + 1.214285715 Combine like terms: 1.785714286 + 1.214285715 = 3.000000001 a = 3.000000001 Simplifying a = 3.000000001Subproblem 2
a + -1.214285715 = -1.785714286 Simplifying a + -1.214285715 = -1.785714286 Reorder the terms: -1.214285715 + a = -1.785714286 Solving -1.214285715 + a = -1.785714286 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.214285715' to each side of the equation. -1.214285715 + 1.214285715 + a = -1.785714286 + 1.214285715 Combine like terms: -1.214285715 + 1.214285715 = 0.000000000 0.000000000 + a = -1.785714286 + 1.214285715 a = -1.785714286 + 1.214285715 Combine like terms: -1.785714286 + 1.214285715 = -0.571428571 a = -0.571428571 Simplifying a = -0.571428571Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.000000001, -0.571428571}
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