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Simplifying 90a2 = -103a + -24 Reorder the terms: 90a2 = -24 + -103a Solving 90a2 = -24 + -103a Solving for variable 'a'. Reorder the terms: 24 + 103a + 90a2 = -24 + -103a + 24 + 103a Reorder the terms: 24 + 103a + 90a2 = -24 + 24 + -103a + 103a Combine like terms: -24 + 24 = 0 24 + 103a + 90a2 = 0 + -103a + 103a 24 + 103a + 90a2 = -103a + 103a Combine like terms: -103a + 103a = 0 24 + 103a + 90a2 = 0 Begin completing the square. Divide all terms by 90 the coefficient of the squared term: Divide each side by '90'. 0.2666666667 + 1.144444444a + a2 = 0 Move the constant term to the right: Add '-0.2666666667' to each side of the equation. 0.2666666667 + 1.144444444a + -0.2666666667 + a2 = 0 + -0.2666666667 Reorder the terms: 0.2666666667 + -0.2666666667 + 1.144444444a + a2 = 0 + -0.2666666667 Combine like terms: 0.2666666667 + -0.2666666667 = 0.0000000000 0.0000000000 + 1.144444444a + a2 = 0 + -0.2666666667 1.144444444a + a2 = 0 + -0.2666666667 Combine like terms: 0 + -0.2666666667 = -0.2666666667 1.144444444a + a2 = -0.2666666667 The a term is 1.144444444a. Take half its coefficient (0.572222222). Square it (0.3274382714) and add it to both sides. Add '0.3274382714' to each side of the equation. 1.144444444a + 0.3274382714 + a2 = -0.2666666667 + 0.3274382714 Reorder the terms: 0.3274382714 + 1.144444444a + a2 = -0.2666666667 + 0.3274382714 Combine like terms: -0.2666666667 + 0.3274382714 = 0.0607716047 0.3274382714 + 1.144444444a + a2 = 0.0607716047 Factor a perfect square on the left side: (a + 0.572222222)(a + 0.572222222) = 0.0607716047 Calculate the square root of the right side: 0.246518974 Break this problem into two subproblems by setting (a + 0.572222222) equal to 0.246518974 and -0.246518974.Subproblem 1
a + 0.572222222 = 0.246518974 Simplifying a + 0.572222222 = 0.246518974 Reorder the terms: 0.572222222 + a = 0.246518974 Solving 0.572222222 + a = 0.246518974 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.572222222' to each side of the equation. 0.572222222 + -0.572222222 + a = 0.246518974 + -0.572222222 Combine like terms: 0.572222222 + -0.572222222 = 0.000000000 0.000000000 + a = 0.246518974 + -0.572222222 a = 0.246518974 + -0.572222222 Combine like terms: 0.246518974 + -0.572222222 = -0.325703248 a = -0.325703248 Simplifying a = -0.325703248Subproblem 2
a + 0.572222222 = -0.246518974 Simplifying a + 0.572222222 = -0.246518974 Reorder the terms: 0.572222222 + a = -0.246518974 Solving 0.572222222 + a = -0.246518974 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.572222222' to each side of the equation. 0.572222222 + -0.572222222 + a = -0.246518974 + -0.572222222 Combine like terms: 0.572222222 + -0.572222222 = 0.000000000 0.000000000 + a = -0.246518974 + -0.572222222 a = -0.246518974 + -0.572222222 Combine like terms: -0.246518974 + -0.572222222 = -0.818741196 a = -0.818741196 Simplifying a = -0.818741196Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.325703248, -0.818741196}
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