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Simplifying 3a2 + -12a + 3 = -1a2 + -4 Reorder the terms: 3 + -12a + 3a2 = -1a2 + -4 Reorder the terms: 3 + -12a + 3a2 = -4 + -1a2 Solving 3 + -12a + 3a2 = -4 + -1a2 Solving for variable 'a'. Reorder the terms: 3 + 4 + -12a + 3a2 + a2 = -4 + -1a2 + 4 + a2 Combine like terms: 3 + 4 = 7 7 + -12a + 3a2 + a2 = -4 + -1a2 + 4 + a2 Combine like terms: 3a2 + a2 = 4a2 7 + -12a + 4a2 = -4 + -1a2 + 4 + a2 Reorder the terms: 7 + -12a + 4a2 = -4 + 4 + -1a2 + a2 Combine like terms: -4 + 4 = 0 7 + -12a + 4a2 = 0 + -1a2 + a2 7 + -12a + 4a2 = -1a2 + a2 Combine like terms: -1a2 + a2 = 0 7 + -12a + 4a2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 1.75 + -3a + a2 = 0 Move the constant term to the right: Add '-1.75' to each side of the equation. 1.75 + -3a + -1.75 + a2 = 0 + -1.75 Reorder the terms: 1.75 + -1.75 + -3a + a2 = 0 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + -3a + a2 = 0 + -1.75 -3a + a2 = 0 + -1.75 Combine like terms: 0 + -1.75 = -1.75 -3a + a2 = -1.75 The a term is -3a. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3a + 2.25 + a2 = -1.75 + 2.25 Reorder the terms: 2.25 + -3a + a2 = -1.75 + 2.25 Combine like terms: -1.75 + 2.25 = 0.5 2.25 + -3a + a2 = 0.5 Factor a perfect square on the left side: (a + -1.5)(a + -1.5) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (a + -1.5) equal to 0.707106781 and -0.707106781.Subproblem 1
a + -1.5 = 0.707106781 Simplifying a + -1.5 = 0.707106781 Reorder the terms: -1.5 + a = 0.707106781 Solving -1.5 + a = 0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = 0.707106781 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = 0.707106781 + 1.5 a = 0.707106781 + 1.5 Combine like terms: 0.707106781 + 1.5 = 2.207106781 a = 2.207106781 Simplifying a = 2.207106781Subproblem 2
a + -1.5 = -0.707106781 Simplifying a + -1.5 = -0.707106781 Reorder the terms: -1.5 + a = -0.707106781 Solving -1.5 + a = -0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + a = -0.707106781 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + a = -0.707106781 + 1.5 a = -0.707106781 + 1.5 Combine like terms: -0.707106781 + 1.5 = 0.792893219 a = 0.792893219 Simplifying a = 0.792893219Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.207106781, 0.792893219}
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