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Simplifying 4 + -2a = 6 + -1(3a + 6) * 2a Reorder the terms: 4 + -2a = 6 + -1(6 + 3a) * 2a Reorder the terms for easier multiplication: 4 + -2a = 6 + -1 * 2a(6 + 3a) Multiply -1 * 2 4 + -2a = 6 + -2a(6 + 3a) 4 + -2a = 6 + (6 * -2a + 3a * -2a) 4 + -2a = 6 + (-12a + -6a2) Solving 4 + -2a = 6 + -12a + -6a2 Solving for variable 'a'. Reorder the terms: 4 + -6 + -2a + 12a + 6a2 = 6 + -12a + -6a2 + -6 + 12a + 6a2 Combine like terms: 4 + -6 = -2 -2 + -2a + 12a + 6a2 = 6 + -12a + -6a2 + -6 + 12a + 6a2 Combine like terms: -2a + 12a = 10a -2 + 10a + 6a2 = 6 + -12a + -6a2 + -6 + 12a + 6a2 Reorder the terms: -2 + 10a + 6a2 = 6 + -6 + -12a + 12a + -6a2 + 6a2 Combine like terms: 6 + -6 = 0 -2 + 10a + 6a2 = 0 + -12a + 12a + -6a2 + 6a2 -2 + 10a + 6a2 = -12a + 12a + -6a2 + 6a2 Combine like terms: -12a + 12a = 0 -2 + 10a + 6a2 = 0 + -6a2 + 6a2 -2 + 10a + 6a2 = -6a2 + 6a2 Combine like terms: -6a2 + 6a2 = 0 -2 + 10a + 6a2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 5a + 3a2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 5a + 3a2)' equal to zero and attempt to solve: Simplifying -1 + 5a + 3a2 = 0 Solving -1 + 5a + 3a2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + 1.666666667a + a2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 1.666666667a + 0.3333333333 + a2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 1.666666667a + a2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 1.666666667a + a2 = 0 + 0.3333333333 1.666666667a + a2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 1.666666667a + a2 = 0.3333333333 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 = 0.3333333333 + 0.6944444447 Reorder the terms: 0.6944444447 + 1.666666667a + a2 = 0.3333333333 + 0.6944444447 Combine like terms: 0.3333333333 + 0.6944444447 = 1.027777778 0.6944444447 + 1.666666667a + a2 = 1.027777778 Factor a perfect square on the left side: (a + 0.8333333335)(a + 0.8333333335) = 1.027777778 Calculate the square root of the right side: 1.013793755 Break this problem into two subproblems by setting (a + 0.8333333335) equal to 1.013793755 and -1.013793755.Subproblem 1
a + 0.8333333335 = 1.013793755 Simplifying a + 0.8333333335 = 1.013793755 Reorder the terms: 0.8333333335 + a = 1.013793755 Solving 0.8333333335 + a = 1.013793755 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 = 1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = 1.013793755 + -0.8333333335 a = 1.013793755 + -0.8333333335 Combine like terms: 1.013793755 + -0.8333333335 = 0.1804604215 a = 0.1804604215 Simplifying a = 0.1804604215Subproblem 2
a + 0.8333333335 = -1.013793755 Simplifying a + 0.8333333335 = -1.013793755 Reorder the terms: 0.8333333335 + a = -1.013793755 Solving 0.8333333335 + a = -1.013793755 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 = -1.013793755 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + a = -1.013793755 + -0.8333333335 a = -1.013793755 + -0.8333333335 Combine like terms: -1.013793755 + -0.8333333335 = -1.8471270885 a = -1.8471270885 Simplifying a = -1.8471270885Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.1804604215, -1.8471270885}Solution
a = {0.1804604215, -1.8471270885}
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