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Simplifying 4a2 + a + -83 = 0 Reorder the terms: -83 + a + 4a2 = 0 Solving -83 + a + 4a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -20.75 + 0.25a + a2 = 0 Move the constant term to the right: Add '20.75' to each side of the equation. -20.75 + 0.25a + 20.75 + a2 = 0 + 20.75 Reorder the terms: -20.75 + 20.75 + 0.25a + a2 = 0 + 20.75 Combine like terms: -20.75 + 20.75 = 0.00 0.00 + 0.25a + a2 = 0 + 20.75 0.25a + a2 = 0 + 20.75 Combine like terms: 0 + 20.75 = 20.75 0.25a + a2 = 20.75 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. 0.25a + 0.25 + a2 = 20.75 + 0.25 Reorder the terms: 0.25 + 0.25a + a2 = 20.75 + 0.25 Combine like terms: 20.75 + 0.25 = 21 0.25 + 0.25a + a2 = 21 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 21 Calculate the square root of the right side: 4.582575695 Break this problem into two subproblems by setting (a + 0.5) equal to 4.582575695 and -4.582575695.Subproblem 1
a + 0.5 = 4.582575695 Simplifying a + 0.5 = 4.582575695 Reorder the terms: 0.5 + a = 4.582575695 Solving 0.5 + a = 4.582575695 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 = 4.582575695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 4.582575695 + -0.5 a = 4.582575695 + -0.5 Combine like terms: 4.582575695 + -0.5 = 4.082575695 a = 4.082575695 Simplifying a = 4.082575695Subproblem 2
a + 0.5 = -4.582575695 Simplifying a + 0.5 = -4.582575695 Reorder the terms: 0.5 + a = -4.582575695 Solving 0.5 + a = -4.582575695 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 = -4.582575695 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -4.582575695 + -0.5 a = -4.582575695 + -0.5 Combine like terms: -4.582575695 + -0.5 = -5.082575695 a = -5.082575695 Simplifying a = -5.082575695Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.082575695, -5.082575695}
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