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Simplifying (5a + -3) + -1(8a + -6) * a = -11 Reorder the terms: (-3 + 5a) + -1(8a + -6) * a = -11 Remove parenthesis around (-3 + 5a) -3 + 5a + -1(8a + -6) * a = -11 Reorder the terms: -3 + 5a + -1(-6 + 8a) * a = -11 Reorder the terms for easier multiplication: -3 + 5a + -1a(-6 + 8a) = -11 -3 + 5a + (-6 * -1a + 8a * -1a) = -11 -3 + 5a + (6a + -8a2) = -11 Combine like terms: 5a + 6a = 11a -3 + 11a + -8a2 = -11 Solving -3 + 11a + -8a2 = -11 Solving for variable 'a'. Reorder the terms: -3 + 11 + 11a + -8a2 = -11 + 11 Combine like terms: -3 + 11 = 8 8 + 11a + -8a2 = -11 + 11 Combine like terms: -11 + 11 = 0 8 + 11a + -8a2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -1 + -1.375a + a2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1.375a + 1 + a2 = 0 + 1 Reorder the terms: -1 + 1 + -1.375a + a2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1.375a + a2 = 0 + 1 -1.375a + a2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1.375a + a2 = 1 The a term is -1.375a. Take half its coefficient (-0.6875). Square it (0.47265625) and add it to both sides. Add '0.47265625' to each side of the equation. -1.375a + 0.47265625 + a2 = 1 + 0.47265625 Reorder the terms: 0.47265625 + -1.375a + a2 = 1 + 0.47265625 Combine like terms: 1 + 0.47265625 = 1.47265625 0.47265625 + -1.375a + a2 = 1.47265625 Factor a perfect square on the left side: (a + -0.6875)(a + -0.6875) = 1.47265625 Calculate the square root of the right side: 1.21353049 Break this problem into two subproblems by setting (a + -0.6875) equal to 1.21353049 and -1.21353049.Subproblem 1
a + -0.6875 = 1.21353049 Simplifying a + -0.6875 = 1.21353049 Reorder the terms: -0.6875 + a = 1.21353049 Solving -0.6875 + a = 1.21353049 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + a = 1.21353049 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + a = 1.21353049 + 0.6875 a = 1.21353049 + 0.6875 Combine like terms: 1.21353049 + 0.6875 = 1.90103049 a = 1.90103049 Simplifying a = 1.90103049Subproblem 2
a + -0.6875 = -1.21353049 Simplifying a + -0.6875 = -1.21353049 Reorder the terms: -0.6875 + a = -1.21353049 Solving -0.6875 + a = -1.21353049 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.6875' to each side of the equation. -0.6875 + 0.6875 + a = -1.21353049 + 0.6875 Combine like terms: -0.6875 + 0.6875 = 0.0000 0.0000 + a = -1.21353049 + 0.6875 a = -1.21353049 + 0.6875 Combine like terms: -1.21353049 + 0.6875 = -0.52603049 a = -0.52603049 Simplifying a = -0.52603049Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.90103049, -0.52603049}
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