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Simplifying 3a2 + ab = b2 Reorder the terms: ab + 3a2 = b2 Solving ab + 3a2 = b2 Solving for variable 'a'. Combine like terms: b2 + -1b2 = 0 ab + 3a2 + -1b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.3333333333ab + a2 + -0.3333333333b2 = 0 Move the constant term to the right: Add '0.3333333333b2' to each side of the equation. 0.3333333333ab + a2 + -0.3333333333b2 + 0.3333333333b2 = 0 + 0.3333333333b2 Combine like terms: -0.3333333333b2 + 0.3333333333b2 = 0.0000000000 0.3333333333ab + a2 + 0.0000000000 = 0 + 0.3333333333b2 0.3333333333ab + a2 = 0 + 0.3333333333b2 Remove the zero: 0.3333333333ab + a2 = 0.3333333333b2 The a term is ab. Take half its coefficient (0.5b). Square it (0.25b2) and add it to both sides. Add '0.25b2' to each side of the equation. 0.3333333333ab + a2 + 0.25b2 = 0.3333333333b2 + 0.25b2 Combine like terms: 0.3333333333b2 + 0.25b2 = 0.5833333333b2 0.3333333333ab + a2 + 0.25b2 = 0.5833333333b2 Factor a perfect square on the left side: (a + 0.5b)(a + 0.5b) = 0.5833333333b2 Calculate the square root of the right side: 0.763762616b Break this problem into two subproblems by setting (a + 0.5b) equal to 0.763762616b and -0.763762616b.Subproblem 1
a + 0.5b = 0.763762616b Simplifying a + 0.5b = 0.763762616b Solving a + 0.5b = 0.763762616b Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5b' to each side of the equation. a + 0.5b + -0.5b = 0.763762616b + -0.5b Combine like terms: 0.5b + -0.5b = 0.0 a + 0.0 = 0.763762616b + -0.5b a = 0.763762616b + -0.5b Combine like terms: 0.763762616b + -0.5b = 0.263762616b a = 0.263762616b Simplifying a = 0.263762616bSubproblem 2
a + 0.5b = -0.763762616b Simplifying a + 0.5b = -0.763762616b Solving a + 0.5b = -0.763762616b Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.5b' to each side of the equation. a + 0.5b + -0.5b = -0.763762616b + -0.5b Combine like terms: 0.5b + -0.5b = 0.0 a + 0.0 = -0.763762616b + -0.5b a = -0.763762616b + -0.5b Combine like terms: -0.763762616b + -0.5b = -1.263762616b a = -1.263762616b Simplifying a = -1.263762616bSolution
The solution to the problem is based on the solutions from the subproblems. a = {0.263762616b, -1.263762616b}
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