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Simplifying 114 = -0.5x2 + 25x + -111 Reorder the terms: 114 = -111 + 25x + -0.5x2 Solving 114 = -111 + 25x + -0.5x2 Solving for variable 'x'. Combine like terms: 114 + 111 = 225 225 + -25x + 0.5x2 = -111 + 25x + -0.5x2 + 111 + -25x + 0.5x2 Reorder the terms: 225 + -25x + 0.5x2 = -111 + 111 + 25x + -25x + -0.5x2 + 0.5x2 Combine like terms: -111 + 111 = 0 225 + -25x + 0.5x2 = 0 + 25x + -25x + -0.5x2 + 0.5x2 225 + -25x + 0.5x2 = 25x + -25x + -0.5x2 + 0.5x2 Combine like terms: 25x + -25x = 0 225 + -25x + 0.5x2 = 0 + -0.5x2 + 0.5x2 225 + -25x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 225 + -25x + 0.5x2 = 0.0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 450 + -50x + x2 = 0 Move the constant term to the right: Add '-450' to each side of the equation. 450 + -50x + -450 + x2 = 0 + -450 Reorder the terms: 450 + -450 + -50x + x2 = 0 + -450 Combine like terms: 450 + -450 = 0 0 + -50x + x2 = 0 + -450 -50x + x2 = 0 + -450 Combine like terms: 0 + -450 = -450 -50x + x2 = -450 The x term is -50x. Take half its coefficient (-25). Square it (625) and add it to both sides. Add '625' to each side of the equation. -50x + 625 + x2 = -450 + 625 Reorder the terms: 625 + -50x + x2 = -450 + 625 Combine like terms: -450 + 625 = 175 625 + -50x + x2 = 175 Factor a perfect square on the left side: (x + -25)(x + -25) = 175 Calculate the square root of the right side: 13.228756555 Break this problem into two subproblems by setting (x + -25) equal to 13.228756555 and -13.228756555.Subproblem 1
x + -25 = 13.228756555 Simplifying x + -25 = 13.228756555 Reorder the terms: -25 + x = 13.228756555 Solving -25 + x = 13.228756555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '25' to each side of the equation. -25 + 25 + x = 13.228756555 + 25 Combine like terms: -25 + 25 = 0 0 + x = 13.228756555 + 25 x = 13.228756555 + 25 Combine like terms: 13.228756555 + 25 = 38.228756555 x = 38.228756555 Simplifying x = 38.228756555Subproblem 2
x + -25 = -13.228756555 Simplifying x + -25 = -13.228756555 Reorder the terms: -25 + x = -13.228756555 Solving -25 + x = -13.228756555 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '25' to each side of the equation. -25 + 25 + x = -13.228756555 + 25 Combine like terms: -25 + 25 = 0 0 + x = -13.228756555 + 25 x = -13.228756555 + 25 Combine like terms: -13.228756555 + 25 = 11.771243445 x = 11.771243445 Simplifying x = 11.771243445Solution
The solution to the problem is based on the solutions from the subproblems. x = {38.228756555, 11.771243445}
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