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Simplifying 2a2 + 3a = 16 Reorder the terms: 3a + 2a2 = 16 Solving 3a + 2a2 = 16 Solving for variable 'a'. Reorder the terms: -16 + 3a + 2a2 = 16 + -16 Combine like terms: 16 + -16 = 0 -16 + 3a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -8 + 1.5a + a2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + 1.5a + 8 + a2 = 0 + 8 Reorder the terms: -8 + 8 + 1.5a + a2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + 1.5a + a2 = 0 + 8 1.5a + a2 = 0 + 8 Combine like terms: 0 + 8 = 8 1.5a + a2 = 8 The a term is 1.5a. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5a + 0.5625 + a2 = 8 + 0.5625 Reorder the terms: 0.5625 + 1.5a + a2 = 8 + 0.5625 Combine like terms: 8 + 0.5625 = 8.5625 0.5625 + 1.5a + a2 = 8.5625 Factor a perfect square on the left side: (a + 0.75)(a + 0.75) = 8.5625 Calculate the square root of the right side: 2.926174978 Break this problem into two subproblems by setting (a + 0.75) equal to 2.926174978 and -2.926174978.Subproblem 1
a + 0.75 = 2.926174978 Simplifying a + 0.75 = 2.926174978 Reorder the terms: 0.75 + a = 2.926174978 Solving 0.75 + a = 2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = 2.926174978 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = 2.926174978 + -0.75 a = 2.926174978 + -0.75 Combine like terms: 2.926174978 + -0.75 = 2.176174978 a = 2.176174978 Simplifying a = 2.176174978Subproblem 2
a + 0.75 = -2.926174978 Simplifying a + 0.75 = -2.926174978 Reorder the terms: 0.75 + a = -2.926174978 Solving 0.75 + a = -2.926174978 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + a = -2.926174978 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + a = -2.926174978 + -0.75 a = -2.926174978 + -0.75 Combine like terms: -2.926174978 + -0.75 = -3.676174978 a = -3.676174978 Simplifying a = -3.676174978Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.176174978, -3.676174978}
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