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Simplifying 0.5a2 + a + -12 = 0 Reorder the terms: -12 + a + 0.5a2 = 0 Solving -12 + a + 0.5a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -24 + 2a + a2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 2a + 24 + a2 = 0 + 24 Reorder the terms: -24 + 24 + 2a + a2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 2a + a2 = 0 + 24 2a + a2 = 0 + 24 Combine like terms: 0 + 24 = 24 2a + a2 = 24 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. 2a + 0.25 + a2 = 24 + 0.25 Reorder the terms: 0.25 + 2a + a2 = 24 + 0.25 Combine like terms: 24 + 0.25 = 24.25 0.25 + 2a + a2 = 24.25 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 24.25 Calculate the square root of the right side: 4.924428901 Break this problem into two subproblems by setting (a + 0.5) equal to 4.924428901 and -4.924428901.Subproblem 1
a + 0.5 = 4.924428901 Simplifying a + 0.5 = 4.924428901 Reorder the terms: 0.5 + a = 4.924428901 Solving 0.5 + a = 4.924428901 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 = 4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.924428901 + -0.5 a = 4.924428901 + -0.5 Combine like terms: 4.924428901 + -0.5 = 4.424428901 a = 4.424428901 Simplifying a = 4.424428901Subproblem 2
a + 0.5 = -4.924428901 Simplifying a + 0.5 = -4.924428901 Reorder the terms: 0.5 + a = -4.924428901 Solving 0.5 + a = -4.924428901 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 = -4.924428901 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.924428901 + -0.5 a = -4.924428901 + -0.5 Combine like terms: -4.924428901 + -0.5 = -5.424428901 a = -5.424428901 Simplifying a = -5.424428901Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.424428901, -5.424428901}
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