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Simplifying 11a2 + 32a + 18 = 20 Reorder the terms: 18 + 32a + 11a2 = 20 Solving 18 + 32a + 11a2 = 20 Solving for variable 'a'. Reorder the terms: 18 + -20 + 32a + 11a2 = 20 + -20 Combine like terms: 18 + -20 = -2 -2 + 32a + 11a2 = 20 + -20 Combine like terms: 20 + -20 = 0 -2 + 32a + 11a2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -0.1818181818 + 2.909090909a + a2 = 0 Move the constant term to the right: Add '0.1818181818' to each side of the equation. -0.1818181818 + 2.909090909a + 0.1818181818 + a2 = 0 + 0.1818181818 Reorder the terms: -0.1818181818 + 0.1818181818 + 2.909090909a + a2 = 0 + 0.1818181818 Combine like terms: -0.1818181818 + 0.1818181818 = 0.0000000000 0.0000000000 + 2.909090909a + a2 = 0 + 0.1818181818 2.909090909a + a2 = 0 + 0.1818181818 Combine like terms: 0 + 0.1818181818 = 0.1818181818 2.909090909a + a2 = 0.1818181818 The a term is 2.909090909a. Take half its coefficient (1.454545455). Square it (2.115702481) and add it to both sides. Add '2.115702481' to each side of the equation. 2.909090909a + 2.115702481 + a2 = 0.1818181818 + 2.115702481 Reorder the terms: 2.115702481 + 2.909090909a + a2 = 0.1818181818 + 2.115702481 Combine like terms: 0.1818181818 + 2.115702481 = 2.2975206628 2.115702481 + 2.909090909a + a2 = 2.2975206628 Factor a perfect square on the left side: (a + 1.454545455)(a + 1.454545455) = 2.2975206628 Calculate the square root of the right side: 1.515757455 Break this problem into two subproblems by setting (a + 1.454545455) equal to 1.515757455 and -1.515757455.Subproblem 1
a + 1.454545455 = 1.515757455 Simplifying a + 1.454545455 = 1.515757455 Reorder the terms: 1.454545455 + a = 1.515757455 Solving 1.454545455 + a = 1.515757455 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.454545455' to each side of the equation. 1.454545455 + -1.454545455 + a = 1.515757455 + -1.454545455 Combine like terms: 1.454545455 + -1.454545455 = 0.000000000 0.000000000 + a = 1.515757455 + -1.454545455 a = 1.515757455 + -1.454545455 Combine like terms: 1.515757455 + -1.454545455 = 0.061212 a = 0.061212 Simplifying a = 0.061212Subproblem 2
a + 1.454545455 = -1.515757455 Simplifying a + 1.454545455 = -1.515757455 Reorder the terms: 1.454545455 + a = -1.515757455 Solving 1.454545455 + a = -1.515757455 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.454545455' to each side of the equation. 1.454545455 + -1.454545455 + a = -1.515757455 + -1.454545455 Combine like terms: 1.454545455 + -1.454545455 = 0.000000000 0.000000000 + a = -1.515757455 + -1.454545455 a = -1.515757455 + -1.454545455 Combine like terms: -1.515757455 + -1.454545455 = -2.97030291 a = -2.97030291 Simplifying a = -2.97030291Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.061212, -2.97030291}
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