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Simplifying 2 = -16x2 + 45x + 2.5 Reorder the terms: 2 = 2.5 + 45x + -16x2 Solving 2 = 2.5 + 45x + -16x2 Solving for variable 'x'. Combine like terms: 2 + -2.5 = -0.5 -0.5 + -45x + 16x2 = 2.5 + 45x + -16x2 + -2.5 + -45x + 16x2 Reorder the terms: -0.5 + -45x + 16x2 = 2.5 + -2.5 + 45x + -45x + -16x2 + 16x2 Combine like terms: 2.5 + -2.5 = 0.0 -0.5 + -45x + 16x2 = 0.0 + 45x + -45x + -16x2 + 16x2 -0.5 + -45x + 16x2 = 45x + -45x + -16x2 + 16x2 Combine like terms: 45x + -45x = 0 -0.5 + -45x + 16x2 = 0 + -16x2 + 16x2 -0.5 + -45x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -0.5 + -45x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.03125 + -2.8125x + x2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + -2.8125x + 0.03125 + x2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + -2.8125x + x2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + -2.8125x + x2 = 0 + 0.03125 -2.8125x + x2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 -2.8125x + x2 = 0.03125 The x term is -2.8125x. Take half its coefficient (-1.40625). Square it (1.977539063) and add it to both sides. Add '1.977539063' to each side of the equation. -2.8125x + 1.977539063 + x2 = 0.03125 + 1.977539063 Reorder the terms: 1.977539063 + -2.8125x + x2 = 0.03125 + 1.977539063 Combine like terms: 0.03125 + 1.977539063 = 2.008789063 1.977539063 + -2.8125x + x2 = 2.008789063 Factor a perfect square on the left side: (x + -1.40625)(x + -1.40625) = 2.008789063 Calculate the square root of the right side: 1.417317559 Break this problem into two subproblems by setting (x + -1.40625) equal to 1.417317559 and -1.417317559.Subproblem 1
x + -1.40625 = 1.417317559 Simplifying x + -1.40625 = 1.417317559 Reorder the terms: -1.40625 + x = 1.417317559 Solving -1.40625 + x = 1.417317559 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + x = 1.417317559 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + x = 1.417317559 + 1.40625 x = 1.417317559 + 1.40625 Combine like terms: 1.417317559 + 1.40625 = 2.823567559 x = 2.823567559 Simplifying x = 2.823567559Subproblem 2
x + -1.40625 = -1.417317559 Simplifying x + -1.40625 = -1.417317559 Reorder the terms: -1.40625 + x = -1.417317559 Solving -1.40625 + x = -1.417317559 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.40625' to each side of the equation. -1.40625 + 1.40625 + x = -1.417317559 + 1.40625 Combine like terms: -1.40625 + 1.40625 = 0.00000 0.00000 + x = -1.417317559 + 1.40625 x = -1.417317559 + 1.40625 Combine like terms: -1.417317559 + 1.40625 = -0.011067559 x = -0.011067559 Simplifying x = -0.011067559Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.823567559, -0.011067559}
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