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Simplifying 7a2 + 16a + -8 = 0 Reorder the terms: -8 + 16a + 7a2 = 0 Solving -8 + 16a + 7a2 = 0 Solving for variable 'a'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.142857143 + 2.285714286a + a2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + 2.285714286a + 1.142857143 + a2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + 2.285714286a + a2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + 2.285714286a + a2 = 0 + 1.142857143 2.285714286a + a2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 2.285714286a + a2 = 1.142857143 The a term is 2.285714286a. Take half its coefficient (1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. 2.285714286a + 1.306122449 + a2 = 1.142857143 + 1.306122449 Reorder the terms: 1.306122449 + 2.285714286a + a2 = 1.142857143 + 1.306122449 Combine like terms: 1.142857143 + 1.306122449 = 2.448979592 1.306122449 + 2.285714286a + a2 = 2.448979592 Factor a perfect square on the left side: (a + 1.142857143)(a + 1.142857143) = 2.448979592 Calculate the square root of the right side: 1.564921593 Break this problem into two subproblems by setting (a + 1.142857143) equal to 1.564921593 and -1.564921593.Subproblem 1
a + 1.142857143 = 1.564921593 Simplifying a + 1.142857143 = 1.564921593 Reorder the terms: 1.142857143 + a = 1.564921593 Solving 1.142857143 + a = 1.564921593 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + a = 1.564921593 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + a = 1.564921593 + -1.142857143 a = 1.564921593 + -1.142857143 Combine like terms: 1.564921593 + -1.142857143 = 0.42206445 a = 0.42206445 Simplifying a = 0.42206445Subproblem 2
a + 1.142857143 = -1.564921593 Simplifying a + 1.142857143 = -1.564921593 Reorder the terms: 1.142857143 + a = -1.564921593 Solving 1.142857143 + a = -1.564921593 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-1.142857143' to each side of the equation. 1.142857143 + -1.142857143 + a = -1.564921593 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + a = -1.564921593 + -1.142857143 a = -1.564921593 + -1.142857143 Combine like terms: -1.564921593 + -1.142857143 = -2.707778736 a = -2.707778736 Simplifying a = -2.707778736Solution
The solution to the problem is based on the solutions from the subproblems. a = {0.42206445, -2.707778736}
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