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Simplifying 3a3 + 5a2 + -6a = 0 Reorder the terms: -6a + 5a2 + 3a3 = 0 Solving -6a + 5a2 + 3a3 = 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'a'. a(-6 + 5a + 3a2) = 0Subproblem 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 '(-6 + 5a + 3a2)' equal to zero and attempt to solve: Simplifying -6 + 5a + 3a2 = 0 Solving -6 + 5a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 1.666666667a + a2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.666666667a + 2 + a2 = 0 + 2 Reorder the terms: -2 + 2 + 1.666666667a + a2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.666666667a + a2 = 0 + 2 1.666666667a + a2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.666666667a + a2 = 2 The a term is 1.666666667a. Take half its coefficient (0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. 1.666666667a + 0.6944444447 + a2 = 2 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667a + a2 = 2 + 0.6944444447 Combine like terms: 2 + 0.6944444447 = 2.6944444447 0.6944444447 + 1.666666667a + a2 = 2.6944444447 Factor a perfect square on the left side: (a + 0.8333333335)(a + 0.8333333335) = 2.6944444447 Calculate the square root of the right side: 1.6414763 Break this problem into two subproblems by setting (a + 0.8333333335) equal to 1.6414763 and -1.6414763.Subproblem 1
a + 0.8333333335 = 1.6414763 Simplifying a + 0.8333333335 = 1.6414763 Reorder the terms: 0.8333333335 + a = 1.6414763 Solving 0.8333333335 + a = 1.6414763 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = 1.6414763 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = 1.6414763 + -0.8333333335 a = 1.6414763 + -0.8333333335 Combine like terms: 1.6414763 + -0.8333333335 = 0.8081429665 a = 0.8081429665 Simplifying a = 0.8081429665Subproblem 2
a + 0.8333333335 = -1.6414763 Simplifying a + 0.8333333335 = -1.6414763 Reorder the terms: 0.8333333335 + a = -1.6414763 Solving 0.8333333335 + a = -1.6414763 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + a = -1.6414763 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = -1.6414763 + -0.8333333335 a = -1.6414763 + -0.8333333335 Combine like terms: -1.6414763 + -0.8333333335 = -2.4748096335 a = -2.4748096335 Simplifying a = -2.4748096335Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.8081429665, -2.4748096335}Solution
a = {0, 0.8081429665, -2.4748096335}
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