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Simplifying 15 + -1(a + 8) * a = 3 Reorder the terms: 15 + -1(8 + a) * a = 3 Reorder the terms for easier multiplication: 15 + -1a(8 + a) = 3 15 + (8 * -1a + a * -1a) = 3 15 + (-8a + -1a2) = 3 Solving 15 + -8a + -1a2 = 3 Solving for variable 'a'. Reorder the terms: 15 + -3 + -8a + -1a2 = 3 + -3 Combine like terms: 15 + -3 = 12 12 + -8a + -1a2 = 3 + -3 Combine like terms: 3 + -3 = 0 12 + -8a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -12 + 8a + a2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 8a + 12 + a2 = 0 + 12 Reorder the terms: -12 + 12 + 8a + a2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 8a + a2 = 0 + 12 8a + a2 = 0 + 12 Combine like terms: 0 + 12 = 12 8a + a2 = 12 The a term is 8a. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8a + 16 + a2 = 12 + 16 Reorder the terms: 16 + 8a + a2 = 12 + 16 Combine like terms: 12 + 16 = 28 16 + 8a + a2 = 28 Factor a perfect square on the left side: (a + 4)(a + 4) = 28 Calculate the square root of the right side: 5.291502622 Break this problem into two subproblems by setting (a + 4) equal to 5.291502622 and -5.291502622.Subproblem 1
a + 4 = 5.291502622 Simplifying a + 4 = 5.291502622 Reorder the terms: 4 + a = 5.291502622 Solving 4 + a = 5.291502622 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + a = 5.291502622 + -4 Combine like terms: 4 + -4 = 0 0 + a = 5.291502622 + -4 a = 5.291502622 + -4 Combine like terms: 5.291502622 + -4 = 1.291502622 a = 1.291502622 Simplifying a = 1.291502622Subproblem 2
a + 4 = -5.291502622 Simplifying a + 4 = -5.291502622 Reorder the terms: 4 + a = -5.291502622 Solving 4 + a = -5.291502622 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + a = -5.291502622 + -4 Combine like terms: 4 + -4 = 0 0 + a = -5.291502622 + -4 a = -5.291502622 + -4 Combine like terms: -5.291502622 + -4 = -9.291502622 a = -9.291502622 Simplifying a = -9.291502622Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.291502622, -9.291502622}
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