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Simplifying 4a2 + -16a = -14 Reorder the terms: -16a + 4a2 = -14 Solving -16a + 4a2 = -14 Solving for variable 'a'. Reorder the terms: 14 + -16a + 4a2 = -14 + 14 Combine like terms: -14 + 14 = 0 14 + -16a + 4a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -8a + 2a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -8a + 2a2)' equal to zero and attempt to solve: Simplifying 7 + -8a + 2a2 = 0 Solving 7 + -8a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 3.5 + -4a + a2 = 0 Move the constant term to the right: Add '-3.5' to each side of the equation. 3.5 + -4a + -3.5 + a2 = 0 + -3.5 Reorder the terms: 3.5 + -3.5 + -4a + a2 = 0 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + -4a + a2 = 0 + -3.5 -4a + a2 = 0 + -3.5 Combine like terms: 0 + -3.5 = -3.5 -4a + a2 = -3.5 The a term is -4a. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4a + 4 + a2 = -3.5 + 4 Reorder the terms: 4 + -4a + a2 = -3.5 + 4 Combine like terms: -3.5 + 4 = 0.5 4 + -4a + a2 = 0.5 Factor a perfect square on the left side: (a + -2)(a + -2) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (a + -2) equal to 0.707106781 and -0.707106781.Subproblem 1
a + -2 = 0.707106781 Simplifying a + -2 = 0.707106781 Reorder the terms: -2 + a = 0.707106781 Solving -2 + a = 0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = 0.707106781 + 2 Combine like terms: -2 + 2 = 0 0 + a = 0.707106781 + 2 a = 0.707106781 + 2 Combine like terms: 0.707106781 + 2 = 2.707106781 a = 2.707106781 Simplifying a = 2.707106781Subproblem 2
a + -2 = -0.707106781 Simplifying a + -2 = -0.707106781 Reorder the terms: -2 + a = -0.707106781 Solving -2 + a = -0.707106781 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + a = -0.707106781 + 2 Combine like terms: -2 + 2 = 0 0 + a = -0.707106781 + 2 a = -0.707106781 + 2 Combine like terms: -0.707106781 + 2 = 1.292893219 a = 1.292893219 Simplifying a = 1.292893219Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.707106781, 1.292893219}Solution
a = {2.707106781, 1.292893219}
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