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Simplifying 7a2 + 7 + 32a = 0 Reorder the terms: 7 + 32a + 7a2 = 0 Solving 7 + 32a + 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 + 4.571428571a + a2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 4.571428571a + -1 + a2 = 0 + -1 Reorder the terms: 1 + -1 + 4.571428571a + a2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 4.571428571a + a2 = 0 + -1 4.571428571a + a2 = 0 + -1 Combine like terms: 0 + -1 = -1 4.571428571a + a2 = -1 The a term is 4.571428571a. Take half its coefficient (2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. 4.571428571a + 5.224489797 + a2 = -1 + 5.224489797 Reorder the terms: 5.224489797 + 4.571428571a + a2 = -1 + 5.224489797 Combine like terms: -1 + 5.224489797 = 4.224489797 5.224489797 + 4.571428571a + a2 = 4.224489797 Factor a perfect square on the left side: (a + 2.285714286)(a + 2.285714286) = 4.224489797 Calculate the square root of the right side: 2.055356367 Break this problem into two subproblems by setting (a + 2.285714286) equal to 2.055356367 and -2.055356367.Subproblem 1
a + 2.285714286 = 2.055356367 Simplifying a + 2.285714286 = 2.055356367 Reorder the terms: 2.285714286 + a = 2.055356367 Solving 2.285714286 + a = 2.055356367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = 2.055356367 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = 2.055356367 + -2.285714286 a = 2.055356367 + -2.285714286 Combine like terms: 2.055356367 + -2.285714286 = -0.230357919 a = -0.230357919 Simplifying a = -0.230357919Subproblem 2
a + 2.285714286 = -2.055356367 Simplifying a + 2.285714286 = -2.055356367 Reorder the terms: 2.285714286 + a = -2.055356367 Solving 2.285714286 + a = -2.055356367 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + a = -2.055356367 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + a = -2.055356367 + -2.285714286 a = -2.055356367 + -2.285714286 Combine like terms: -2.055356367 + -2.285714286 = -4.341070653 a = -4.341070653 Simplifying a = -4.341070653Solution
The solution to the problem is based on the solutions from the subproblems. a = {-0.230357919, -4.341070653}
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