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Simplifying 20000 = -0.5x2 + 222x + -1600 Reorder the terms: 20000 = -1600 + 222x + -0.5x2 Solving 20000 = -1600 + 222x + -0.5x2 Solving for variable 'x'. Combine like terms: 20000 + 1600 = 21600 21600 + -222x + 0.5x2 = -1600 + 222x + -0.5x2 + 1600 + -222x + 0.5x2 Reorder the terms: 21600 + -222x + 0.5x2 = -1600 + 1600 + 222x + -222x + -0.5x2 + 0.5x2 Combine like terms: -1600 + 1600 = 0 21600 + -222x + 0.5x2 = 0 + 222x + -222x + -0.5x2 + 0.5x2 21600 + -222x + 0.5x2 = 222x + -222x + -0.5x2 + 0.5x2 Combine like terms: 222x + -222x = 0 21600 + -222x + 0.5x2 = 0 + -0.5x2 + 0.5x2 21600 + -222x + 0.5x2 = -0.5x2 + 0.5x2 Combine like terms: -0.5x2 + 0.5x2 = 0.0 21600 + -222x + 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'. 43200 + -444x + x2 = 0 Move the constant term to the right: Add '-43200' to each side of the equation. 43200 + -444x + -43200 + x2 = 0 + -43200 Reorder the terms: 43200 + -43200 + -444x + x2 = 0 + -43200 Combine like terms: 43200 + -43200 = 0 0 + -444x + x2 = 0 + -43200 -444x + x2 = 0 + -43200 Combine like terms: 0 + -43200 = -43200 -444x + x2 = -43200 The x term is -444x. Take half its coefficient (-222). Square it (49284) and add it to both sides. Add '49284' to each side of the equation. -444x + 49284 + x2 = -43200 + 49284 Reorder the terms: 49284 + -444x + x2 = -43200 + 49284 Combine like terms: -43200 + 49284 = 6084 49284 + -444x + x2 = 6084 Factor a perfect square on the left side: (x + -222)(x + -222) = 6084 Calculate the square root of the right side: 78 Break this problem into two subproblems by setting (x + -222) equal to 78 and -78.Subproblem 1
x + -222 = 78 Simplifying x + -222 = 78 Reorder the terms: -222 + x = 78 Solving -222 + x = 78 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '222' to each side of the equation. -222 + 222 + x = 78 + 222 Combine like terms: -222 + 222 = 0 0 + x = 78 + 222 x = 78 + 222 Combine like terms: 78 + 222 = 300 x = 300 Simplifying x = 300Subproblem 2
x + -222 = -78 Simplifying x + -222 = -78 Reorder the terms: -222 + x = -78 Solving -222 + x = -78 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '222' to each side of the equation. -222 + 222 + x = -78 + 222 Combine like terms: -222 + 222 = 0 0 + x = -78 + 222 x = -78 + 222 Combine like terms: -78 + 222 = 144 x = 144 Simplifying x = 144Solution
The solution to the problem is based on the solutions from the subproblems. x = {300, 144}
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