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Simplifying 230 = a2 + 3a Reorder the terms: 230 = 3a + a2 Solving 230 = 3a + a2 Solving for variable 'a'. Reorder the terms: 230 + -3a + -1a2 = 3a + -3a + a2 + -1a2 Combine like terms: 3a + -3a = 0 230 + -3a + -1a2 = 0 + a2 + -1a2 230 + -3a + -1a2 = a2 + -1a2 Combine like terms: a2 + -1a2 = 0 230 + -3a + -1a2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -230 + 3a + a2 = 0 Move the constant term to the right: Add '230' to each side of the equation. -230 + 3a + 230 + a2 = 0 + 230 Reorder the terms: -230 + 230 + 3a + a2 = 0 + 230 Combine like terms: -230 + 230 = 0 0 + 3a + a2 = 0 + 230 3a + a2 = 0 + 230 Combine like terms: 0 + 230 = 230 3a + a2 = 230 The a term is 3a. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3a + 2.25 + a2 = 230 + 2.25 Reorder the terms: 2.25 + 3a + a2 = 230 + 2.25 Combine like terms: 230 + 2.25 = 232.25 2.25 + 3a + a2 = 232.25 Factor a perfect square on the left side: (a + 1.5)(a + 1.5) = 232.25 Calculate the square root of the right side: 15.239750654 Break this problem into two subproblems by setting (a + 1.5) equal to 15.239750654 and -15.239750654.Subproblem 1
a + 1.5 = 15.239750654 Simplifying a + 1.5 = 15.239750654 Reorder the terms: 1.5 + a = 15.239750654 Solving 1.5 + a = 15.239750654 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = 15.239750654 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = 15.239750654 + -1.5 a = 15.239750654 + -1.5 Combine like terms: 15.239750654 + -1.5 = 13.739750654 a = 13.739750654 Simplifying a = 13.739750654Subproblem 2
a + 1.5 = -15.239750654 Simplifying a + 1.5 = -15.239750654 Reorder the terms: 1.5 + a = -15.239750654 Solving 1.5 + a = -15.239750654 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + a = -15.239750654 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + a = -15.239750654 + -1.5 a = -15.239750654 + -1.5 Combine like terms: -15.239750654 + -1.5 = -16.739750654 a = -16.739750654 Simplifying a = -16.739750654Solution
The solution to the problem is based on the solutions from the subproblems. a = {13.739750654, -16.739750654}
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