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Simplifying a(14a + 5) = 12 Reorder the terms: a(5 + 14a) = 12 (5 * a + 14a * a) = 12 (5a + 14a2) = 12 Solving 5a + 14a2 = 12 Solving for variable 'a'. Reorder the terms: -12 + 5a + 14a2 = 12 + -12 Combine like terms: 12 + -12 = 0 -12 + 5a + 14a2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.8571428571 + 0.3571428571a + a2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + 0.3571428571a + 0.8571428571 + a2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + 0.3571428571a + a2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + 0.3571428571a + a2 = 0 + 0.8571428571 0.3571428571a + a2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 0.3571428571a + a2 = 0.8571428571 The a term is 0.3571428571a. Take half its coefficient (0.1785714286). Square it (0.03188775511) and add it to both sides. Add '0.03188775511' to each side of the equation. 0.3571428571a + 0.03188775511 + a2 = 0.8571428571 + 0.03188775511 Reorder the terms: 0.03188775511 + 0.3571428571a + a2 = 0.8571428571 + 0.03188775511 Combine like terms: 0.8571428571 + 0.03188775511 = 0.88903061221 0.03188775511 + 0.3571428571a + a2 = 0.88903061221 Factor a perfect square on the left side: (a + 0.1785714286)(a + 0.1785714286) = 0.88903061221 Calculate the square root of the right side: 0.942884199 Break this problem into two subproblems by setting (a + 0.1785714286) equal to 0.942884199 and -0.942884199.Subproblem 1
a + 0.1785714286 = 0.942884199 Simplifying a + 0.1785714286 = 0.942884199 Reorder the terms: 0.1785714286 + a = 0.942884199 Solving 0.1785714286 + a = 0.942884199 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1785714286' to each side of the equation. 0.1785714286 + -0.1785714286 + a = 0.942884199 + -0.1785714286 Combine like terms: 0.1785714286 + -0.1785714286 = 0.0000000000 0.0000000000 + a = 0.942884199 + -0.1785714286 a = 0.942884199 + -0.1785714286 Combine like terms: 0.942884199 + -0.1785714286 = 0.7643127704 a = 0.7643127704 Simplifying a = 0.7643127704Subproblem 2
a + 0.1785714286 = -0.942884199 Simplifying a + 0.1785714286 = -0.942884199 Reorder the terms: 0.1785714286 + a = -0.942884199 Solving 0.1785714286 + a = -0.942884199 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.1785714286' to each side of the equation. 0.1785714286 + -0.1785714286 + a = -0.942884199 + -0.1785714286 Combine like terms: 0.1785714286 + -0.1785714286 = 0.0000000000 0.0000000000 + a = -0.942884199 + -0.1785714286 a = -0.942884199 + -0.1785714286 Combine like terms: -0.942884199 + -0.1785714286 = -1.1214556276 a = -1.1214556276 Simplifying a = -1.1214556276Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.7643127704, -1.1214556276}
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