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Simplifying 8a(2a2 + -5a + -6) + -2a(a + -3) = 0 Reorder the terms: 8a(-6 + -5a + 2a2) + -2a(a + -3) = 0 (-6 * 8a + -5a * 8a + 2a2 * 8a) + -2a(a + -3) = 0 (-48a + -40a2 + 16a3) + -2a(a + -3) = 0 Reorder the terms: -48a + -40a2 + 16a3 + -2a(-3 + a) = 0 -48a + -40a2 + 16a3 + (-3 * -2a + a * -2a) = 0 -48a + -40a2 + 16a3 + (6a + -2a2) = 0 Reorder the terms: -48a + 6a + -40a2 + -2a2 + 16a3 = 0 Combine like terms: -48a + 6a = -42a -42a + -40a2 + -2a2 + 16a3 = 0 Combine like terms: -40a2 + -2a2 = -42a2 -42a + -42a2 + 16a3 = 0 Solving -42a + -42a2 + 16a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), '2a'. 2a(-21 + -21a + 8a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'a' equal to zero and attempt to solve: Simplifying a = 0 Solving a = 0 Move all terms containing a to the left, all other terms to the right. Simplifying a = 0Subproblem 2
Set the factor '(-21 + -21a + 8a2)' equal to zero and attempt to solve: Simplifying -21 + -21a + 8a2 = 0 Solving -21 + -21a + 8a2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.625 + -2.625a + a2 = 0 Move the constant term to the right: Add '2.625' to each side of the equation. -2.625 + -2.625a + 2.625 + a2 = 0 + 2.625 Reorder the terms: -2.625 + 2.625 + -2.625a + a2 = 0 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + -2.625a + a2 = 0 + 2.625 -2.625a + a2 = 0 + 2.625 Combine like terms: 0 + 2.625 = 2.625 -2.625a + a2 = 2.625 The a term is -2.625a. Take half its coefficient (-1.3125). Square it (1.72265625) and add it to both sides. Add '1.72265625' to each side of the equation. -2.625a + 1.72265625 + a2 = 2.625 + 1.72265625 Reorder the terms: 1.72265625 + -2.625a + a2 = 2.625 + 1.72265625 Combine like terms: 2.625 + 1.72265625 = 4.34765625 1.72265625 + -2.625a + a2 = 4.34765625 Factor a perfect square on the left side: (a + -1.3125)(a + -1.3125) = 4.34765625 Calculate the square root of the right side: 2.085103415 Break this problem into two subproblems by setting (a + -1.3125) equal to 2.085103415 and -2.085103415.Subproblem 1
a + -1.3125 = 2.085103415 Simplifying a + -1.3125 = 2.085103415 Reorder the terms: -1.3125 + a = 2.085103415 Solving -1.3125 + a = 2.085103415 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + a = 2.085103415 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + a = 2.085103415 + 1.3125 a = 2.085103415 + 1.3125 Combine like terms: 2.085103415 + 1.3125 = 3.397603415 a = 3.397603415 Simplifying a = 3.397603415Subproblem 2
a + -1.3125 = -2.085103415 Simplifying a + -1.3125 = -2.085103415 Reorder the terms: -1.3125 + a = -2.085103415 Solving -1.3125 + a = -2.085103415 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '1.3125' to each side of the equation. -1.3125 + 1.3125 + a = -2.085103415 + 1.3125 Combine like terms: -1.3125 + 1.3125 = 0.0000 0.0000 + a = -2.085103415 + 1.3125 a = -2.085103415 + 1.3125 Combine like terms: -2.085103415 + 1.3125 = -0.772603415 a = -0.772603415 Simplifying a = -0.772603415Solution
The solution to the problem is based on the solutions from the subproblems. a = {3.397603415, -0.772603415}Solution
a = {0, 3.397603415, -0.772603415}
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