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Simplifying 2.5 = -16x2 + 33x + 46 Reorder the terms: 2.5 = 46 + 33x + -16x2 Solving 2.5 = 46 + 33x + -16x2 Solving for variable 'x'. Combine like terms: 2.5 + -46 = -43.5 -43.5 + -33x + 16x2 = 46 + 33x + -16x2 + -46 + -33x + 16x2 Reorder the terms: -43.5 + -33x + 16x2 = 46 + -46 + 33x + -33x + -16x2 + 16x2 Combine like terms: 46 + -46 = 0 -43.5 + -33x + 16x2 = 0 + 33x + -33x + -16x2 + 16x2 -43.5 + -33x + 16x2 = 33x + -33x + -16x2 + 16x2 Combine like terms: 33x + -33x = 0 -43.5 + -33x + 16x2 = 0 + -16x2 + 16x2 -43.5 + -33x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -43.5 + -33x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -2.71875 + -2.0625x + x2 = 0 Move the constant term to the right: Add '2.71875' to each side of the equation. -2.71875 + -2.0625x + 2.71875 + x2 = 0 + 2.71875 Reorder the terms: -2.71875 + 2.71875 + -2.0625x + x2 = 0 + 2.71875 Combine like terms: -2.71875 + 2.71875 = 0.00000 0.00000 + -2.0625x + x2 = 0 + 2.71875 -2.0625x + x2 = 0 + 2.71875 Combine like terms: 0 + 2.71875 = 2.71875 -2.0625x + x2 = 2.71875 The x term is -2.0625x. Take half its coefficient (-1.03125). Square it (1.063476563) and add it to both sides. Add '1.063476563' to each side of the equation. -2.0625x + 1.063476563 + x2 = 2.71875 + 1.063476563 Reorder the terms: 1.063476563 + -2.0625x + x2 = 2.71875 + 1.063476563 Combine like terms: 2.71875 + 1.063476563 = 3.782226563 1.063476563 + -2.0625x + x2 = 3.782226563 Factor a perfect square on the left side: (x + -1.03125)(x + -1.03125) = 3.782226563 Calculate the square root of the right side: 1.944794735 Break this problem into two subproblems by setting (x + -1.03125) equal to 1.944794735 and -1.944794735.Subproblem 1
x + -1.03125 = 1.944794735 Simplifying x + -1.03125 = 1.944794735 Reorder the terms: -1.03125 + x = 1.944794735 Solving -1.03125 + x = 1.944794735 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + x = 1.944794735 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + x = 1.944794735 + 1.03125 x = 1.944794735 + 1.03125 Combine like terms: 1.944794735 + 1.03125 = 2.976044735 x = 2.976044735 Simplifying x = 2.976044735Subproblem 2
x + -1.03125 = -1.944794735 Simplifying x + -1.03125 = -1.944794735 Reorder the terms: -1.03125 + x = -1.944794735 Solving -1.03125 + x = -1.944794735 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.03125' to each side of the equation. -1.03125 + 1.03125 + x = -1.944794735 + 1.03125 Combine like terms: -1.03125 + 1.03125 = 0.00000 0.00000 + x = -1.944794735 + 1.03125 x = -1.944794735 + 1.03125 Combine like terms: -1.944794735 + 1.03125 = -0.913544735 x = -0.913544735 Simplifying x = -0.913544735Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.976044735, -0.913544735}
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