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Simplifying 2a2 = 6 + 8a Solving 2a2 = 6 + 8a Solving for variable 'a'. Reorder the terms: -6 + -8a + 2a2 = 6 + 8a + -6 + -8a Reorder the terms: -6 + -8a + 2a2 = 6 + -6 + 8a + -8a Combine like terms: 6 + -6 = 0 -6 + -8a + 2a2 = 0 + 8a + -8a -6 + -8a + 2a2 = 8a + -8a Combine like terms: 8a + -8a = 0 -6 + -8a + 2a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-3 + -4a + a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-3 + -4a + a2)' equal to zero and attempt to solve: Simplifying -3 + -4a + a2 = 0 Solving -3 + -4a + a2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + -4a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + -4a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + -4a + a2 = 0 + 3 -4a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 -4a + a2 = 3 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 + 4 Reorder the terms: 4 + -4a + a2 = 3 + 4 Combine like terms: 3 + 4 = 7 4 + -4a + a2 = 7 Factor a perfect square on the left side: (a + -2)(a + -2) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (a + -2) equal to 2.645751311 and -2.645751311.Subproblem 1
a + -2 = 2.645751311 Simplifying a + -2 = 2.645751311 Reorder the terms: -2 + a = 2.645751311 Solving -2 + a = 2.645751311 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 = 2.645751311 + 2 Combine like terms: -2 + 2 = 0 0 + a = 2.645751311 + 2 a = 2.645751311 + 2 Combine like terms: 2.645751311 + 2 = 4.645751311 a = 4.645751311 Simplifying a = 4.645751311Subproblem 2
a + -2 = -2.645751311 Simplifying a + -2 = -2.645751311 Reorder the terms: -2 + a = -2.645751311 Solving -2 + a = -2.645751311 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 = -2.645751311 + 2 Combine like terms: -2 + 2 = 0 0 + a = -2.645751311 + 2 a = -2.645751311 + 2 Combine like terms: -2.645751311 + 2 = -0.645751311 a = -0.645751311 Simplifying a = -0.645751311Solution
The solution to the problem is based on the solutions from the subproblems. a = {4.645751311, -0.645751311}Solution
a = {4.645751311, -0.645751311}
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