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Simplifying (5a2 + -2a + -8) + -1(a2 + -6a + 3) = 0 Reorder the terms: (-8 + -2a + 5a2) + -1(a2 + -6a + 3) = 0 Remove parenthesis around (-8 + -2a + 5a2) -8 + -2a + 5a2 + -1(a2 + -6a + 3) = 0 Reorder the terms: -8 + -2a + 5a2 + -1(3 + -6a + a2) = 0 -8 + -2a + 5a2 + (3 * -1 + -6a * -1 + a2 * -1) = 0 -8 + -2a + 5a2 + (-3 + 6a + -1a2) = 0 Reorder the terms: -8 + -3 + -2a + 6a + 5a2 + -1a2 = 0 Combine like terms: -8 + -3 = -11 -11 + -2a + 6a + 5a2 + -1a2 = 0 Combine like terms: -2a + 6a = 4a -11 + 4a + 5a2 + -1a2 = 0 Combine like terms: 5a2 + -1a2 = 4a2 -11 + 4a + 4a2 = 0 Solving -11 + 4a + 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'. -2.75 + a + a2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + a + 2.75 + a2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + a + a2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + a + a2 = 0 + 2.75 a + a2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 a + a2 = 2.75 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. + 0.25 + a2 = 2.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + a2 = 2.75 + 0.25 Combine like terms: 2.75 + 0.25 = 3 1.25 + a2 = 3 Factor a perfect square on the left side: (a + 0.5)(a + 0.5) = 3 Calculate the square root of the right side: 1.732050808 Break this problem into two subproblems by setting (a + 0.5) equal to 1.732050808 and -1.732050808.Subproblem 1
a + 0.5 = 1.732050808 Simplifying a + 0.5 = 1.732050808 Reorder the terms: 0.5 + a = 1.732050808 Solving 0.5 + a = 1.732050808 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 = 1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = 1.732050808 + -0.5 a = 1.732050808 + -0.5 Combine like terms: 1.732050808 + -0.5 = 1.232050808 a = 1.232050808 Simplifying a = 1.232050808Subproblem 2
a + 0.5 = -1.732050808 Simplifying a + 0.5 = -1.732050808 Reorder the terms: 0.5 + a = -1.732050808 Solving 0.5 + a = -1.732050808 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 = -1.732050808 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + a = -1.732050808 + -0.5 a = -1.732050808 + -0.5 Combine like terms: -1.732050808 + -0.5 = -2.232050808 a = -2.232050808 Simplifying a = -2.232050808Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.232050808, -2.232050808}
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