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Simplifying 16x2 + -41x + -32 = 0 Reorder the terms: -32 + -41x + 16x2 = 0 Solving -32 + -41x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2 + -2.5625x + x2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -2.5625x + 2 + x2 = 0 + 2 Reorder the terms: -2 + 2 + -2.5625x + x2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -2.5625x + x2 = 0 + 2 -2.5625x + x2 = 0 + 2 Combine like terms: 0 + 2 = 2 -2.5625x + x2 = 2 The x term is -2.5625x. Take half its coefficient (-1.28125). Square it (1.641601563) and add it to both sides. Add '1.641601563' to each side of the equation. -2.5625x + 1.641601563 + x2 = 2 + 1.641601563 Reorder the terms: 1.641601563 + -2.5625x + x2 = 2 + 1.641601563 Combine like terms: 2 + 1.641601563 = 3.641601563 1.641601563 + -2.5625x + x2 = 3.641601563 Factor a perfect square on the left side: (x + -1.28125)(x + -1.28125) = 3.641601563 Calculate the square root of the right side: 1.90829808 Break this problem into two subproblems by setting (x + -1.28125) equal to 1.90829808 and -1.90829808.Subproblem 1
x + -1.28125 = 1.90829808 Simplifying x + -1.28125 = 1.90829808 Reorder the terms: -1.28125 + x = 1.90829808 Solving -1.28125 + x = 1.90829808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + x = 1.90829808 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + x = 1.90829808 + 1.28125 x = 1.90829808 + 1.28125 Combine like terms: 1.90829808 + 1.28125 = 3.18954808 x = 3.18954808 Simplifying x = 3.18954808Subproblem 2
x + -1.28125 = -1.90829808 Simplifying x + -1.28125 = -1.90829808 Reorder the terms: -1.28125 + x = -1.90829808 Solving -1.28125 + x = -1.90829808 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.28125' to each side of the equation. -1.28125 + 1.28125 + x = -1.90829808 + 1.28125 Combine like terms: -1.28125 + 1.28125 = 0.00000 0.00000 + x = -1.90829808 + 1.28125 x = -1.90829808 + 1.28125 Combine like terms: -1.90829808 + 1.28125 = -0.62704808 x = -0.62704808 Simplifying x = -0.62704808Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.18954808, -0.62704808}
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