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Simplifying (8a2 + -6a + -8) + -1(-6a2 + 7a + 5) = 0 Reorder the terms: (-8 + -6a + 8a2) + -1(-6a2 + 7a + 5) = 0 Remove parenthesis around (-8 + -6a + 8a2) -8 + -6a + 8a2 + -1(-6a2 + 7a + 5) = 0 Reorder the terms: -8 + -6a + 8a2 + -1(5 + 7a + -6a2) = 0 -8 + -6a + 8a2 + (5 * -1 + 7a * -1 + -6a2 * -1) = 0 -8 + -6a + 8a2 + (-5 + -7a + 6a2) = 0 Reorder the terms: -8 + -5 + -6a + -7a + 8a2 + 6a2 = 0 Combine like terms: -8 + -5 = -13 -13 + -6a + -7a + 8a2 + 6a2 = 0 Combine like terms: -6a + -7a = -13a -13 + -13a + 8a2 + 6a2 = 0 Combine like terms: 8a2 + 6a2 = 14a2 -13 + -13a + 14a2 = 0 Solving -13 + -13a + 14a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.9285714286 + -0.9285714286a + a2 = 0 Move the constant term to the right: Add '0.9285714286' to each side of the equation. -0.9285714286 + -0.9285714286a + 0.9285714286 + a2 = 0 + 0.9285714286 Reorder the terms: -0.9285714286 + 0.9285714286 + -0.9285714286a + a2 = 0 + 0.9285714286 Combine like terms: -0.9285714286 + 0.9285714286 = 0.0000000000 0.0000000000 + -0.9285714286a + a2 = 0 + 0.9285714286 -0.9285714286a + a2 = 0 + 0.9285714286 Combine like terms: 0 + 0.9285714286 = 0.9285714286 -0.9285714286a + a2 = 0.9285714286 The a term is -0.9285714286a. Take half its coefficient (-0.4642857143). Square it (0.2155612245) and add it to both sides. Add '0.2155612245' to each side of the equation. -0.9285714286a + 0.2155612245 + a2 = 0.9285714286 + 0.2155612245 Reorder the terms: 0.2155612245 + -0.9285714286a + a2 = 0.9285714286 + 0.2155612245 Combine like terms: 0.9285714286 + 0.2155612245 = 1.1441326531 0.2155612245 + -0.9285714286a + a2 = 1.1441326531 Factor a perfect square on the left side: (a + -0.4642857143)(a + -0.4642857143) = 1.1441326531 Calculate the square root of the right side: 1.069641367 Break this problem into two subproblems by setting (a + -0.4642857143) equal to 1.069641367 and -1.069641367.Subproblem 1
a + -0.4642857143 = 1.069641367 Simplifying a + -0.4642857143 = 1.069641367 Reorder the terms: -0.4642857143 + a = 1.069641367 Solving -0.4642857143 + a = 1.069641367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4642857143' to each side of the equation. -0.4642857143 + 0.4642857143 + a = 1.069641367 + 0.4642857143 Combine like terms: -0.4642857143 + 0.4642857143 = 0.0000000000 0.0000000000 + a = 1.069641367 + 0.4642857143 a = 1.069641367 + 0.4642857143 Combine like terms: 1.069641367 + 0.4642857143 = 1.5339270813 a = 1.5339270813 Simplifying a = 1.5339270813Subproblem 2
a + -0.4642857143 = -1.069641367 Simplifying a + -0.4642857143 = -1.069641367 Reorder the terms: -0.4642857143 + a = -1.069641367 Solving -0.4642857143 + a = -1.069641367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '0.4642857143' to each side of the equation. -0.4642857143 + 0.4642857143 + a = -1.069641367 + 0.4642857143 Combine like terms: -0.4642857143 + 0.4642857143 = 0.0000000000 0.0000000000 + a = -1.069641367 + 0.4642857143 a = -1.069641367 + 0.4642857143 Combine like terms: -1.069641367 + 0.4642857143 = -0.6053556527 a = -0.6053556527 Simplifying a = -0.6053556527Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.5339270813, -0.6053556527}
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