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Simplifying 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 1 * 2 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 2 * 3 6 * 4 * 5 * 6 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 6 * 4 24 * 5 * 6 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 24 * 5 120 * 6 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 120 * 6 720 * 7 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 720 * 7 5040 * 8 * 9 * 10a = 68a * a + 45a + 12 Multiply 5040 * 8 40320 * 9 * 10a = 68a * a + 45a + 12 Multiply 40320 * 9 362880 * 10a = 68a * a + 45a + 12 Multiply 362880 * 10 3628800a = 68a * a + 45a + 12 Multiply a * a 3628800a = 68a2 + 45a + 12 Reorder the terms: 3628800a = 12 + 45a + 68a2 Solving 3628800a = 12 + 45a + 68a2 Solving for variable 'a'. Reorder the terms: -12 + 3628800a + -45a + -68a2 = 12 + 45a + 68a2 + -12 + -45a + -68a2 Combine like terms: 3628800a + -45a = 3628755a -12 + 3628755a + -68a2 = 12 + 45a + 68a2 + -12 + -45a + -68a2 Reorder the terms: -12 + 3628755a + -68a2 = 12 + -12 + 45a + -45a + 68a2 + -68a2 Combine like terms: 12 + -12 = 0 -12 + 3628755a + -68a2 = 0 + 45a + -45a + 68a2 + -68a2 -12 + 3628755a + -68a2 = 45a + -45a + 68a2 + -68a2 Combine like terms: 45a + -45a = 0 -12 + 3628755a + -68a2 = 0 + 68a2 + -68a2 -12 + 3628755a + -68a2 = 68a2 + -68a2 Combine like terms: 68a2 + -68a2 = 0 -12 + 3628755a + -68a2 = 0 Begin completing the square. Divide all terms by -68 the coefficient of the squared term: Divide each side by '-68'. 0.1764705882 + -53364.04412a + a2 = 0 Move the constant term to the right: Add '-0.1764705882' to each side of the equation. 0.1764705882 + -53364.04412a + -0.1764705882 + a2 = 0 + -0.1764705882 Reorder the terms: 0.1764705882 + -0.1764705882 + -53364.04412a + a2 = 0 + -0.1764705882 Combine like terms: 0.1764705882 + -0.1764705882 = 0.0000000000 0.0000000000 + -53364.04412a + a2 = 0 + -0.1764705882 -53364.04412a + a2 = 0 + -0.1764705882 Combine like terms: 0 + -0.1764705882 = -0.1764705882 -53364.04412a + a2 = -0.1764705882 The a term is -53364.04412a. Take half its coefficient (-26682.02206). Square it (711930301.2) and add it to both sides. Add '711930301.2' to each side of the equation. -53364.04412a + 711930301.2 + a2 = -0.1764705882 + 711930301.2 Reorder the terms: 711930301.2 + -53364.04412a + a2 = -0.1764705882 + 711930301.2 Combine like terms: -0.1764705882 + 711930301.2 = 711930301.0235294118 711930301.2 + -53364.04412a + a2 = 711930301.0235294118 Factor a perfect square on the left side: (a + -26682.02206)(a + -26682.02206) = 711930301.0235294118 Calculate the square root of the right side: 26682.0220565 Break this problem into two subproblems by setting (a + -26682.02206) equal to 26682.0220565 and -26682.0220565.Subproblem 1
a + -26682.02206 = 26682.0220565 Simplifying a + -26682.02206 = 26682.0220565 Reorder the terms: -26682.02206 + a = 26682.0220565 Solving -26682.02206 + a = 26682.0220565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '26682.02206' to each side of the equation. -26682.02206 + 26682.02206 + a = 26682.0220565 + 26682.02206 Combine like terms: -26682.02206 + 26682.02206 = 0.00000 0.00000 + a = 26682.0220565 + 26682.02206 a = 26682.0220565 + 26682.02206 Combine like terms: 26682.0220565 + 26682.02206 = 53364.0441165 a = 53364.0441165 Simplifying a = 53364.0441165Subproblem 2
a + -26682.02206 = -26682.0220565 Simplifying a + -26682.02206 = -26682.0220565 Reorder the terms: -26682.02206 + a = -26682.0220565 Solving -26682.02206 + a = -26682.0220565 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '26682.02206' to each side of the equation. -26682.02206 + 26682.02206 + a = -26682.0220565 + 26682.02206 Combine like terms: -26682.02206 + 26682.02206 = 0.00000 0.00000 + a = -26682.0220565 + 26682.02206 a = -26682.0220565 + 26682.02206 Combine like terms: -26682.0220565 + 26682.02206 = 0.0000035 a = 0.0000035 Simplifying a = 0.0000035Solution
The solution to the problem is based on the solutions from the subproblems. a = {53364.0441165, 0.0000035}
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