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Simplifying 24 = a + 2a * a Multiply a * a 24 = a + 2a2 Solving 24 = a + 2a2 Solving for variable 'a'. Reorder the terms: 24 + -1a + -2a2 = a + -1a + 2a2 + -2a2 Combine like terms: a + -1a = 0 24 + -1a + -2a2 = 0 + 2a2 + -2a2 24 + -1a + -2a2 = 2a2 + -2a2 Combine like terms: 2a2 + -2a2 = 0 24 + -1a + -2a2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12 + 0.5a + a2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 0.5a + 12 + a2 = 0 + 12 Reorder the terms: -12 + 12 + 0.5a + a2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 0.5a + a2 = 0 + 12 0.5a + a2 = 0 + 12 Combine like terms: 0 + 12 = 12 0.5a + a2 = 12 The a term is 0.5a. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5a + 0.0625 + a2 = 12 + 0.0625 Reorder the terms: 0.0625 + 0.5a + a2 = 12 + 0.0625 Combine like terms: 12 + 0.0625 = 12.0625 0.0625 + 0.5a + a2 = 12.0625 Factor a perfect square on the left side: (a + 0.25)(a + 0.25) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (a + 0.25) equal to 3.473110997 and -3.473110997.Subproblem 1
a + 0.25 = 3.473110997 Simplifying a + 0.25 = 3.473110997 Reorder the terms: 0.25 + a = 3.473110997 Solving 0.25 + a = 3.473110997 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = 3.473110997 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = 3.473110997 + -0.25 a = 3.473110997 + -0.25 Combine like terms: 3.473110997 + -0.25 = 3.223110997 a = 3.223110997 Simplifying a = 3.223110997Subproblem 2
a + 0.25 = -3.473110997 Simplifying a + 0.25 = -3.473110997 Reorder the terms: 0.25 + a = -3.473110997 Solving 0.25 + a = -3.473110997 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + a = -3.473110997 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + a = -3.473110997 + -0.25 a = -3.473110997 + -0.25 Combine like terms: -3.473110997 + -0.25 = -3.723110997 a = -3.723110997 Simplifying a = -3.723110997Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.223110997, -3.723110997}
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