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Simplifying 12a2 = 6a + 24 Reorder the terms: 12a2 = 24 + 6a Solving 12a2 = 24 + 6a Solving for variable 'a'. Reorder the terms: -24 + -6a + 12a2 = 24 + 6a + -24 + -6a Reorder the terms: -24 + -6a + 12a2 = 24 + -24 + 6a + -6a Combine like terms: 24 + -24 = 0 -24 + -6a + 12a2 = 0 + 6a + -6a -24 + -6a + 12a2 = 6a + -6a Combine like terms: 6a + -6a = 0 -24 + -6a + 12a2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-4 + -1a + 2a2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-4 + -1a + 2a2)' equal to zero and attempt to solve: Simplifying -4 + -1a + 2a2 = 0 Solving -4 + -1a + 2a2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + -0.5a + a2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -0.5a + 2 + a2 = 0 + 2 Reorder the terms: -2 + 2 + -0.5a + a2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -0.5a + a2 = 0 + 2 -0.5a + a2 = 0 + 2 Combine like terms: 0 + 2 = 2 -0.5a + a2 = 2 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 = 2 + 0.0625 Reorder the terms: 0.0625 + -0.5a + a2 = 2 + 0.0625 Combine like terms: 2 + 0.0625 = 2.0625 0.0625 + -0.5a + a2 = 2.0625 Factor a perfect square on the left side: (a + -0.25)(a + -0.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (a + -0.25) equal to 1.436140662 and -1.436140662.Subproblem 1
a + -0.25 = 1.436140662 Simplifying a + -0.25 = 1.436140662 Reorder the terms: -0.25 + a = 1.436140662 Solving -0.25 + a = 1.436140662 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 = 1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = 1.436140662 + 0.25 a = 1.436140662 + 0.25 Combine like terms: 1.436140662 + 0.25 = 1.686140662 a = 1.686140662 Simplifying a = 1.686140662Subproblem 2
a + -0.25 = -1.436140662 Simplifying a + -0.25 = -1.436140662 Reorder the terms: -0.25 + a = -1.436140662 Solving -0.25 + a = -1.436140662 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 = -1.436140662 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + a = -1.436140662 + 0.25 a = -1.436140662 + 0.25 Combine like terms: -1.436140662 + 0.25 = -1.186140662 a = -1.186140662 Simplifying a = -1.186140662Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.686140662, -1.186140662}Solution
a = {1.686140662, -1.186140662}
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