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Simplifying 7a2 + -4a + 32a = 9 + 12a2 Reorder the terms: -4a + 32a + 7a2 = 9 + 12a2 Combine like terms: -4a + 32a = 28a 28a + 7a2 = 9 + 12a2 Solving 28a + 7a2 = 9 + 12a2 Solving for variable 'a'. Reorder the terms: -9 + 28a + 7a2 + -12a2 = 9 + 12a2 + -9 + -12a2 Combine like terms: 7a2 + -12a2 = -5a2 -9 + 28a + -5a2 = 9 + 12a2 + -9 + -12a2 Reorder the terms: -9 + 28a + -5a2 = 9 + -9 + 12a2 + -12a2 Combine like terms: 9 + -9 = 0 -9 + 28a + -5a2 = 0 + 12a2 + -12a2 -9 + 28a + -5a2 = 12a2 + -12a2 Combine like terms: 12a2 + -12a2 = 0 -9 + 28a + -5a2 = 0 Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. 1.8 + -5.6a + a2 = 0 Move the constant term to the right: Add '-1.8' to each side of the equation. 1.8 + -5.6a + -1.8 + a2 = 0 + -1.8 Reorder the terms: 1.8 + -1.8 + -5.6a + a2 = 0 + -1.8 Combine like terms: 1.8 + -1.8 = 0.0 0.0 + -5.6a + a2 = 0 + -1.8 -5.6a + a2 = 0 + -1.8 Combine like terms: 0 + -1.8 = -1.8 -5.6a + a2 = -1.8 The a term is -5.6a. Take half its coefficient (-2.8). Square it (7.84) and add it to both sides. Add '7.84' to each side of the equation. -5.6a + 7.84 + a2 = -1.8 + 7.84 Reorder the terms: 7.84 + -5.6a + a2 = -1.8 + 7.84 Combine like terms: -1.8 + 7.84 = 6.04 7.84 + -5.6a + a2 = 6.04 Factor a perfect square on the left side: (a + -2.8)(a + -2.8) = 6.04 Calculate the square root of the right side: 2.457641145 Break this problem into two subproblems by setting (a + -2.8) equal to 2.457641145 and -2.457641145.Subproblem 1
a + -2.8 = 2.457641145 Simplifying a + -2.8 = 2.457641145 Reorder the terms: -2.8 + a = 2.457641145 Solving -2.8 + a = 2.457641145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.8' to each side of the equation. -2.8 + 2.8 + a = 2.457641145 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + a = 2.457641145 + 2.8 a = 2.457641145 + 2.8 Combine like terms: 2.457641145 + 2.8 = 5.257641145 a = 5.257641145 Simplifying a = 5.257641145Subproblem 2
a + -2.8 = -2.457641145 Simplifying a + -2.8 = -2.457641145 Reorder the terms: -2.8 + a = -2.457641145 Solving -2.8 + a = -2.457641145 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '2.8' to each side of the equation. -2.8 + 2.8 + a = -2.457641145 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + a = -2.457641145 + 2.8 a = -2.457641145 + 2.8 Combine like terms: -2.457641145 + 2.8 = 0.342358855 a = 0.342358855 Simplifying a = 0.342358855Solution
The solution to the problem is based on the solutions from the subproblems. a = {5.257641145, 0.342358855}
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