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Simplifying 7a2 + 6a + -19 = 2 Reorder the terms: -19 + 6a + 7a2 = 2 Solving -19 + 6a + 7a2 = 2 Solving for variable 'a'. Reorder the terms: -19 + -2 + 6a + 7a2 = 2 + -2 Combine like terms: -19 + -2 = -21 -21 + 6a + 7a2 = 2 + -2 Combine like terms: 2 + -2 = 0 -21 + 6a + 7a2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -3 + 0.8571428571a + a2 = 0 Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.8571428571a + 3 + a2 = 0 + 3 Reorder the terms: -3 + 3 + 0.8571428571a + a2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.8571428571a + a2 = 0 + 3 0.8571428571a + a2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.8571428571a + a2 = 3 The a term is 0.8571428571a. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571a + 0.1836734694 + a2 = 3 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571a + a2 = 3 + 0.1836734694 Combine like terms: 3 + 0.1836734694 = 3.1836734694 0.1836734694 + 0.8571428571a + a2 = 3.1836734694 Factor a perfect square on the left side: (a + 0.4285714286)(a + 0.4285714286) = 3.1836734694 Calculate the square root of the right side: 1.784285142 Break this problem into two subproblems by setting (a + 0.4285714286) equal to 1.784285142 and -1.784285142.Subproblem 1
a + 0.4285714286 = 1.784285142 Simplifying a + 0.4285714286 = 1.784285142 Reorder the terms: 0.4285714286 + a = 1.784285142 Solving 0.4285714286 + a = 1.784285142 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = 1.784285142 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = 1.784285142 + -0.4285714286 a = 1.784285142 + -0.4285714286 Combine like terms: 1.784285142 + -0.4285714286 = 1.3557137134 a = 1.3557137134 Simplifying a = 1.3557137134Subproblem 2
a + 0.4285714286 = -1.784285142 Simplifying a + 0.4285714286 = -1.784285142 Reorder the terms: 0.4285714286 + a = -1.784285142 Solving 0.4285714286 + a = -1.784285142 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + a = -1.784285142 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + a = -1.784285142 + -0.4285714286 a = -1.784285142 + -0.4285714286 Combine like terms: -1.784285142 + -0.4285714286 = -2.2128565706 a = -2.2128565706 Simplifying a = -2.2128565706Solution
The solution to the problem is based on the solutions from the subproblems. a = {1.3557137134, -2.2128565706}
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