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Simplifying 9a2 + 48a + 64 = 32 Reorder the terms: 64 + 48a + 9a2 = 32 Solving 64 + 48a + 9a2 = 32 Solving for variable 'a'. Reorder the terms: 64 + -32 + 48a + 9a2 = 32 + -32 Combine like terms: 64 + -32 = 32 32 + 48a + 9a2 = 32 + -32 Combine like terms: 32 + -32 = 0 32 + 48a + 9a2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 3.555555556 + 5.333333333a + a2 = 0 Move the constant term to the right: Add '-3.555555556' to each side of the equation. 3.555555556 + 5.333333333a + -3.555555556 + a2 = 0 + -3.555555556 Reorder the terms: 3.555555556 + -3.555555556 + 5.333333333a + a2 = 0 + -3.555555556 Combine like terms: 3.555555556 + -3.555555556 = 0.000000000 0.000000000 + 5.333333333a + a2 = 0 + -3.555555556 5.333333333a + a2 = 0 + -3.555555556 Combine like terms: 0 + -3.555555556 = -3.555555556 5.333333333a + a2 = -3.555555556 The a term is 5.333333333a. Take half its coefficient (2.666666667). Square it (7.111111113) and add it to both sides. Add '7.111111113' to each side of the equation. 5.333333333a + 7.111111113 + a2 = -3.555555556 + 7.111111113 Reorder the terms: 7.111111113 + 5.333333333a + a2 = -3.555555556 + 7.111111113 Combine like terms: -3.555555556 + 7.111111113 = 3.555555557 7.111111113 + 5.333333333a + a2 = 3.555555557 Factor a perfect square on the left side: (a + 2.666666667)(a + 2.666666667) = 3.555555557 Calculate the square root of the right side: 1.885618084 Break this problem into two subproblems by setting (a + 2.666666667) equal to 1.885618084 and -1.885618084.Subproblem 1
a + 2.666666667 = 1.885618084 Simplifying a + 2.666666667 = 1.885618084 Reorder the terms: 2.666666667 + a = 1.885618084 Solving 2.666666667 + a = 1.885618084 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + a = 1.885618084 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + a = 1.885618084 + -2.666666667 a = 1.885618084 + -2.666666667 Combine like terms: 1.885618084 + -2.666666667 = -0.781048583 a = -0.781048583 Simplifying a = -0.781048583Subproblem 2
a + 2.666666667 = -1.885618084 Simplifying a + 2.666666667 = -1.885618084 Reorder the terms: 2.666666667 + a = -1.885618084 Solving 2.666666667 + a = -1.885618084 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.666666667' to each side of the equation. 2.666666667 + -2.666666667 + a = -1.885618084 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + a = -1.885618084 + -2.666666667 a = -1.885618084 + -2.666666667 Combine like terms: -1.885618084 + -2.666666667 = -4.552284751 a = -4.552284751 Simplifying a = -4.552284751Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.781048583, -4.552284751}
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