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Simplifying -7a2 + 32a + 306 = 0 Reorder the terms: 306 + 32a + -7a2 = 0 Solving 306 + 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'. -43.71428571 + -4.571428571a + a2 = 0 Move the constant term to the right: Add '43.71428571' to each side of the equation. -43.71428571 + -4.571428571a + 43.71428571 + a2 = 0 + 43.71428571 Reorder the terms: -43.71428571 + 43.71428571 + -4.571428571a + a2 = 0 + 43.71428571 Combine like terms: -43.71428571 + 43.71428571 = 0.00000000 0.00000000 + -4.571428571a + a2 = 0 + 43.71428571 -4.571428571a + a2 = 0 + 43.71428571 Combine like terms: 0 + 43.71428571 = 43.71428571 -4.571428571a + a2 = 43.71428571 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 = 43.71428571 + 5.224489797 Reorder the terms: 5.224489797 + -4.571428571a + a2 = 43.71428571 + 5.224489797 Combine like terms: 43.71428571 + 5.224489797 = 48.938775507 5.224489797 + -4.571428571a + a2 = 48.938775507 Factor a perfect square on the left side: (a + -2.285714286)(a + -2.285714286) = 48.938775507 Calculate the square root of the right side: 6.995625455 Break this problem into two subproblems by setting (a + -2.285714286) equal to 6.995625455 and -6.995625455.Subproblem 1
a + -2.285714286 = 6.995625455 Simplifying a + -2.285714286 = 6.995625455 Reorder the terms: -2.285714286 + a = 6.995625455 Solving -2.285714286 + a = 6.995625455 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 = 6.995625455 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + a = 6.995625455 + 2.285714286 a = 6.995625455 + 2.285714286 Combine like terms: 6.995625455 + 2.285714286 = 9.281339741 a = 9.281339741 Simplifying a = 9.281339741Subproblem 2
a + -2.285714286 = -6.995625455 Simplifying a + -2.285714286 = -6.995625455 Reorder the terms: -2.285714286 + a = -6.995625455 Solving -2.285714286 + a = -6.995625455 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 = -6.995625455 + 2.285714286 Combine like terms: -2.285714286 + 2.285714286 = 0.000000000 0.000000000 + a = -6.995625455 + 2.285714286 a = -6.995625455 + 2.285714286 Combine like terms: -6.995625455 + 2.285714286 = -4.709911169 a = -4.709911169 Simplifying a = -4.709911169Solution
The solution to the problem is based on the solutions from the subproblems. a = {9.281339741, -4.709911169}
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