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Simplifying 5a2 + -2a = 23 Reorder the terms: -2a + 5a2 = 23 Solving -2a + 5a2 = 23 Solving for variable 'a'. Reorder the terms: -23 + -2a + 5a2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -2a + 5a2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -4.6 + -0.4a + a2 = 0 Move the constant term to the right: Add '4.6' to each side of the equation. -4.6 + -0.4a + 4.6 + a2 = 0 + 4.6 Reorder the terms: -4.6 + 4.6 + -0.4a + a2 = 0 + 4.6 Combine like terms: -4.6 + 4.6 = 0.0 0.0 + -0.4a + a2 = 0 + 4.6 -0.4a + a2 = 0 + 4.6 Combine like terms: 0 + 4.6 = 4.6 -0.4a + a2 = 4.6 The a term is -0.4a. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4a + 0.04 + a2 = 4.6 + 0.04 Reorder the terms: 0.04 + -0.4a + a2 = 4.6 + 0.04 Combine like terms: 4.6 + 0.04 = 4.64 0.04 + -0.4a + a2 = 4.64 Factor a perfect square on the left side: (a + -0.2)(a + -0.2) = 4.64 Calculate the square root of the right side: 2.154065923 Break this problem into two subproblems by setting (a + -0.2) equal to 2.154065923 and -2.154065923.Subproblem 1
a + -0.2 = 2.154065923 Simplifying a + -0.2 = 2.154065923 Reorder the terms: -0.2 + a = 2.154065923 Solving -0.2 + a = 2.154065923 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + a = 2.154065923 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + a = 2.154065923 + 0.2 a = 2.154065923 + 0.2 Combine like terms: 2.154065923 + 0.2 = 2.354065923 a = 2.354065923 Simplifying a = 2.354065923Subproblem 2
a + -0.2 = -2.154065923 Simplifying a + -0.2 = -2.154065923 Reorder the terms: -0.2 + a = -2.154065923 Solving -0.2 + a = -2.154065923 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + a = -2.154065923 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + a = -2.154065923 + 0.2 a = -2.154065923 + 0.2 Combine like terms: -2.154065923 + 0.2 = -1.954065923 a = -1.954065923 Simplifying a = -1.954065923Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.354065923, -1.954065923}
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