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Simplifying 2a2 + a = 186 Reorder the terms: a + 2a2 = 186 Solving a + 2a2 = 186 Solving for variable 'a'. Reorder the terms: -186 + a + 2a2 = 186 + -186 Combine like terms: 186 + -186 = 0 -186 + a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -93 + 0.5a + a2 = 0 Move the constant term to the right: Add '93' to each side of the equation. -93 + 0.5a + 93 + a2 = 0 + 93 Reorder the terms: -93 + 93 + 0.5a + a2 = 0 + 93 Combine like terms: -93 + 93 = 0 0 + 0.5a + a2 = 0 + 93 0.5a + a2 = 0 + 93 Combine like terms: 0 + 93 = 93 0.5a + a2 = 93 The a term is a. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5a + 0.25 + a2 = 93 + 0.25 Reorder the terms: 0.25 + 0.5a + a2 = 93 + 0.25 Combine like terms: 93 + 0.25 = 93.25 0.25 + 0.5a + a2 = 93.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 93.25 Calculate the square root of the right side: 9.656603958 Break this problem into two subproblems by setting (a + 0.5) equal to 9.656603958 and -9.656603958.Subproblem 1
a + 0.5 = 9.656603958 Simplifying a + 0.5 = 9.656603958 Reorder the terms: 0.5 + a = 9.656603958 Solving 0.5 + a = 9.656603958 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = 9.656603958 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 9.656603958 + -0.5 a = 9.656603958 + -0.5 Combine like terms: 9.656603958 + -0.5 = 9.156603958 a = 9.156603958 Simplifying a = 9.156603958Subproblem 2
a + 0.5 = -9.656603958 Simplifying a + 0.5 = -9.656603958 Reorder the terms: 0.5 + a = -9.656603958 Solving 0.5 + a = -9.656603958 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + a = -9.656603958 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -9.656603958 + -0.5 a = -9.656603958 + -0.5 Combine like terms: -9.656603958 + -0.5 = -10.156603958 a = -10.156603958 Simplifying a = -10.156603958Solution
The solution to the problem is based on the solutions from the subproblems. a = {9.156603958, -10.156603958}
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