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Simplifying 19a + -3a(a + -6) = 9 Reorder the terms: 19a + -3a(-6 + a) = 9 19a + (-6 * -3a + a * -3a) = 9 19a + (18a + -3a2) = 9 Combine like terms: 19a + 18a = 37a 37a + -3a2 = 9 Solving 37a + -3a2 = 9 Solving for variable 'a'. Reorder the terms: -9 + 37a + -3a2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 37a + -3a2 = 0 Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. 3 + -12.33333333a + a2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -12.33333333a + -3 + a2 = 0 + -3 Reorder the terms: 3 + -3 + -12.33333333a + a2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -12.33333333a + a2 = 0 + -3 -12.33333333a + a2 = 0 + -3 Combine like terms: 0 + -3 = -3 -12.33333333a + a2 = -3 The a term is -12.33333333a. Take half its coefficient (-6.166666665). Square it (38.02777776) and add it to both sides. Add '38.02777776' to each side of the equation. -12.33333333a + 38.02777776 + a2 = -3 + 38.02777776 Reorder the terms: 38.02777776 + -12.33333333a + a2 = -3 + 38.02777776 Combine like terms: -3 + 38.02777776 = 35.02777776 38.02777776 + -12.33333333a + a2 = 35.02777776 Factor a perfect square on the left side: (a + -6.166666665)(a + -6.166666665) = 35.02777776 Calculate the square root of the right side: 5.918426967 Break this problem into two subproblems by setting (a + -6.166666665) equal to 5.918426967 and -5.918426967.Subproblem 1
a + -6.166666665 = 5.918426967 Simplifying a + -6.166666665 = 5.918426967 Reorder the terms: -6.166666665 + a = 5.918426967 Solving -6.166666665 + a = 5.918426967 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + a = 5.918426967 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + a = 5.918426967 + 6.166666665 a = 5.918426967 + 6.166666665 Combine like terms: 5.918426967 + 6.166666665 = 12.085093632 a = 12.085093632 Simplifying a = 12.085093632Subproblem 2
a + -6.166666665 = -5.918426967 Simplifying a + -6.166666665 = -5.918426967 Reorder the terms: -6.166666665 + a = -5.918426967 Solving -6.166666665 + a = -5.918426967 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + a = -5.918426967 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + a = -5.918426967 + 6.166666665 a = -5.918426967 + 6.166666665 Combine like terms: -5.918426967 + 6.166666665 = 0.248239698 a = 0.248239698 Simplifying a = 0.248239698Solution
The solution to the problem is based on the solutions from the subproblems. a = {12.085093632, 0.248239698}
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