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Simplifying 7(a + 8) * a = 32 Reorder the terms: 7(8 + a) * a = 32 Reorder the terms for easier multiplication: 7a(8 + a) = 32 (8 * 7a + a * 7a) = 32 (56a + 7a2) = 32 Solving 56a + 7a2 = 32 Solving for variable 'a'. Reorder the terms: -32 + 56a + 7a2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 56a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -4.571428571 + 8a + a2 = 0 Move the constant term to the right: Add '4.571428571' to each side of the equation. -4.571428571 + 8a + 4.571428571 + a2 = 0 + 4.571428571 Reorder the terms: -4.571428571 + 4.571428571 + 8a + a2 = 0 + 4.571428571 Combine like terms: -4.571428571 + 4.571428571 = 0.000000000 0.000000000 + 8a + a2 = 0 + 4.571428571 8a + a2 = 0 + 4.571428571 Combine like terms: 0 + 4.571428571 = 4.571428571 8a + a2 = 4.571428571 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 = 4.571428571 + 16 Reorder the terms: 16 + 8a + a2 = 4.571428571 + 16 Combine like terms: 4.571428571 + 16 = 20.571428571 16 + 8a + a2 = 20.571428571 Factor a perfect square on the left side: (a + 4)(a + 4) = 20.571428571 Calculate the square root of the right side: 4.535573676 Break this problem into two subproblems by setting (a + 4) equal to 4.535573676 and -4.535573676.Subproblem 1
a + 4 = 4.535573676 Simplifying a + 4 = 4.535573676 Reorder the terms: 4 + a = 4.535573676 Solving 4 + a = 4.535573676 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 = 4.535573676 + -4 Combine like terms: 4 + -4 = 0 0 + a = 4.535573676 + -4 a = 4.535573676 + -4 Combine like terms: 4.535573676 + -4 = 0.535573676 a = 0.535573676 Simplifying a = 0.535573676Subproblem 2
a + 4 = -4.535573676 Simplifying a + 4 = -4.535573676 Reorder the terms: 4 + a = -4.535573676 Solving 4 + a = -4.535573676 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 = -4.535573676 + -4 Combine like terms: 4 + -4 = 0 0 + a = -4.535573676 + -4 a = -4.535573676 + -4 Combine like terms: -4.535573676 + -4 = -8.535573676 a = -8.535573676 Simplifying a = -8.535573676Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.535573676, -8.535573676}
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