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Simplifying 6a3 + 7a2 = 2a Reorder the terms: 7a2 + 6a3 = 2a Solving 7a2 + 6a3 = 2a Solving for variable 'a'. Reorder the terms: -2a + 7a2 + 6a3 = 2a + -2a Combine like terms: 2a + -2a = 0 -2a + 7a2 + 6a3 = 0 Factor out the Greatest Common Factor (GCF), 'a'. a(-2 + 7a + 6a2) = 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 '(-2 + 7a + 6a2)' equal to zero and attempt to solve: Simplifying -2 + 7a + 6a2 = 0 Solving -2 + 7a + 6a2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.3333333333 + 1.166666667a + a2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 1.166666667a + 0.3333333333 + a2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 1.166666667a + a2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 1.166666667a + a2 = 0 + 0.3333333333 1.166666667a + a2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 1.166666667a + a2 = 0.3333333333 The a term is 1.166666667a. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667a + 0.3402777780 + a2 = 0.3333333333 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667a + a2 = 0.3333333333 + 0.3402777780 Combine like terms: 0.3333333333 + 0.3402777780 = 0.6736111113 0.3402777780 + 1.166666667a + a2 = 0.6736111113 Factor a perfect square on the left side: (a + 0.5833333335)(a + 0.5833333335) = 0.6736111113 Calculate the square root of the right side: 0.82073815 Break this problem into two subproblems by setting (a + 0.5833333335) equal to 0.82073815 and -0.82073815.Subproblem 1
a + 0.5833333335 = 0.82073815 Simplifying a + 0.5833333335 = 0.82073815 Reorder the terms: 0.5833333335 + a = 0.82073815 Solving 0.5833333335 + a = 0.82073815 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + a = 0.82073815 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + a = 0.82073815 + -0.5833333335 a = 0.82073815 + -0.5833333335 Combine like terms: 0.82073815 + -0.5833333335 = 0.2374048165 a = 0.2374048165 Simplifying a = 0.2374048165Subproblem 2
a + 0.5833333335 = -0.82073815 Simplifying a + 0.5833333335 = -0.82073815 Reorder the terms: 0.5833333335 + a = -0.82073815 Solving 0.5833333335 + a = -0.82073815 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + a = -0.82073815 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + a = -0.82073815 + -0.5833333335 a = -0.82073815 + -0.5833333335 Combine like terms: -0.82073815 + -0.5833333335 = -1.4040714835 a = -1.4040714835 Simplifying a = -1.4040714835Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.2374048165, -1.4040714835}Solution
a = {0, 0.2374048165, -1.4040714835}
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