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Simplifying (9a + -8)(5a + 6) = 180 Reorder the terms: (-8 + 9a)(5a + 6) = 180 Reorder the terms: (-8 + 9a)(6 + 5a) = 180 Multiply (-8 + 9a) * (6 + 5a) (-8(6 + 5a) + 9a * (6 + 5a)) = 180 ((6 * -8 + 5a * -8) + 9a * (6 + 5a)) = 180 ((-48 + -40a) + 9a * (6 + 5a)) = 180 (-48 + -40a + (6 * 9a + 5a * 9a)) = 180 (-48 + -40a + (54a + 45a2)) = 180 Combine like terms: -40a + 54a = 14a (-48 + 14a + 45a2) = 180 Solving -48 + 14a + 45a2 = 180 Solving for variable 'a'. Reorder the terms: -48 + -180 + 14a + 45a2 = 180 + -180 Combine like terms: -48 + -180 = -228 -228 + 14a + 45a2 = 180 + -180 Combine like terms: 180 + -180 = 0 -228 + 14a + 45a2 = 0 Begin completing the square. Divide all terms by 45 the coefficient of the squared term: Divide each side by '45'. -5.066666667 + 0.3111111111a + a2 = 0 Move the constant term to the right: Add '5.066666667' to each side of the equation. -5.066666667 + 0.3111111111a + 5.066666667 + a2 = 0 + 5.066666667 Reorder the terms: -5.066666667 + 5.066666667 + 0.3111111111a + a2 = 0 + 5.066666667 Combine like terms: -5.066666667 + 5.066666667 = 0.000000000 0.000000000 + 0.3111111111a + a2 = 0 + 5.066666667 0.3111111111a + a2 = 0 + 5.066666667 Combine like terms: 0 + 5.066666667 = 5.066666667 0.3111111111a + a2 = 5.066666667 The a term is 0.3111111111a. Take half its coefficient (0.1555555556). Square it (0.02419753088) and add it to both sides. Add '0.02419753088' to each side of the equation. 0.3111111111a + 0.02419753088 + a2 = 5.066666667 + 0.02419753088 Reorder the terms: 0.02419753088 + 0.3111111111a + a2 = 5.066666667 + 0.02419753088 Combine like terms: 5.066666667 + 0.02419753088 = 5.09086419788 0.02419753088 + 0.3111111111a + a2 = 5.09086419788 Factor a perfect square on the left side: (a + 0.1555555556)(a + 0.1555555556) = 5.09086419788 Calculate the square root of the right side: 2.256294351 Break this problem into two subproblems by setting (a + 0.1555555556) equal to 2.256294351 and -2.256294351.Subproblem 1
a + 0.1555555556 = 2.256294351 Simplifying a + 0.1555555556 = 2.256294351 Reorder the terms: 0.1555555556 + a = 2.256294351 Solving 0.1555555556 + a = 2.256294351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1555555556' to each side of the equation. 0.1555555556 + -0.1555555556 + a = 2.256294351 + -0.1555555556 Combine like terms: 0.1555555556 + -0.1555555556 = 0.0000000000 0.0000000000 + a = 2.256294351 + -0.1555555556 a = 2.256294351 + -0.1555555556 Combine like terms: 2.256294351 + -0.1555555556 = 2.1007387954 a = 2.1007387954 Simplifying a = 2.1007387954Subproblem 2
a + 0.1555555556 = -2.256294351 Simplifying a + 0.1555555556 = -2.256294351 Reorder the terms: 0.1555555556 + a = -2.256294351 Solving 0.1555555556 + a = -2.256294351 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1555555556' to each side of the equation. 0.1555555556 + -0.1555555556 + a = -2.256294351 + -0.1555555556 Combine like terms: 0.1555555556 + -0.1555555556 = 0.0000000000 0.0000000000 + a = -2.256294351 + -0.1555555556 a = -2.256294351 + -0.1555555556 Combine like terms: -2.256294351 + -0.1555555556 = -2.4118499066 a = -2.4118499066 Simplifying a = -2.4118499066Solution
The solution to the problem is based on the solutions from the subproblems. a = {2.1007387954, -2.4118499066}
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