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Simplifying 18 + 11a = -5a(-7a + -3) Reorder the terms: 18 + 11a = -5a(-3 + -7a) 18 + 11a = (-3 * -5a + -7a * -5a) 18 + 11a = (15a + 35a2) Solving 18 + 11a = 15a + 35a2 Solving for variable 'a'. Combine like terms: 11a + -15a = -4a 18 + -4a + -35a2 = 15a + 35a2 + -15a + -35a2 Reorder the terms: 18 + -4a + -35a2 = 15a + -15a + 35a2 + -35a2 Combine like terms: 15a + -15a = 0 18 + -4a + -35a2 = 0 + 35a2 + -35a2 18 + -4a + -35a2 = 35a2 + -35a2 Combine like terms: 35a2 + -35a2 = 0 18 + -4a + -35a2 = 0 Begin completing the square. Divide all terms by -35 the coefficient of the squared term: Divide each side by '-35'. -0.5142857143 + 0.1142857143a + a2 = 0 Move the constant term to the right: Add '0.5142857143' to each side of the equation. -0.5142857143 + 0.1142857143a + 0.5142857143 + a2 = 0 + 0.5142857143 Reorder the terms: -0.5142857143 + 0.5142857143 + 0.1142857143a + a2 = 0 + 0.5142857143 Combine like terms: -0.5142857143 + 0.5142857143 = 0.0000000000 0.0000000000 + 0.1142857143a + a2 = 0 + 0.5142857143 0.1142857143a + a2 = 0 + 0.5142857143 Combine like terms: 0 + 0.5142857143 = 0.5142857143 0.1142857143a + a2 = 0.5142857143 The a term is 0.1142857143a. Take half its coefficient (0.05714285715). Square it (0.003265306123) and add it to both sides. Add '0.003265306123' to each side of the equation. 0.1142857143a + 0.003265306123 + a2 = 0.5142857143 + 0.003265306123 Reorder the terms: 0.003265306123 + 0.1142857143a + a2 = 0.5142857143 + 0.003265306123 Combine like terms: 0.5142857143 + 0.003265306123 = 0.517551020423 0.003265306123 + 0.1142857143a + a2 = 0.517551020423 Factor a perfect square on the left side: (a + 0.05714285715)(a + 0.05714285715) = 0.517551020423 Calculate the square root of the right side: 0.719410189 Break this problem into two subproblems by setting (a + 0.05714285715) equal to 0.719410189 and -0.719410189.Subproblem 1
a + 0.05714285715 = 0.719410189 Simplifying a + 0.05714285715 = 0.719410189 Reorder the terms: 0.05714285715 + a = 0.719410189 Solving 0.05714285715 + a = 0.719410189 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.05714285715' to each side of the equation. 0.05714285715 + -0.05714285715 + a = 0.719410189 + -0.05714285715 Combine like terms: 0.05714285715 + -0.05714285715 = 0.00000000000 0.00000000000 + a = 0.719410189 + -0.05714285715 a = 0.719410189 + -0.05714285715 Combine like terms: 0.719410189 + -0.05714285715 = 0.66226733185 a = 0.66226733185 Simplifying a = 0.66226733185Subproblem 2
a + 0.05714285715 = -0.719410189 Simplifying a + 0.05714285715 = -0.719410189 Reorder the terms: 0.05714285715 + a = -0.719410189 Solving 0.05714285715 + a = -0.719410189 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.05714285715' to each side of the equation. 0.05714285715 + -0.05714285715 + a = -0.719410189 + -0.05714285715 Combine like terms: 0.05714285715 + -0.05714285715 = 0.00000000000 0.00000000000 + a = -0.719410189 + -0.05714285715 a = -0.719410189 + -0.05714285715 Combine like terms: -0.719410189 + -0.05714285715 = -0.77655304615 a = -0.77655304615 Simplifying a = -0.77655304615Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.66226733185, -0.77655304615}
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