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Simplifying 0.5x2 + x + -120 = 0 Reorder the terms: -120 + x + 0.5x2 = 0 Solving -120 + x + 0.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -240 + 2x + x2 = 0 Move the constant term to the right: Add '240' to each side of the equation. -240 + 2x + 240 + x2 = 0 + 240 Reorder the terms: -240 + 240 + 2x + x2 = 0 + 240 Combine like terms: -240 + 240 = 0 0 + 2x + x2 = 0 + 240 2x + x2 = 0 + 240 Combine like terms: 0 + 240 = 240 2x + x2 = 240 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 2x + 0.25 + x2 = 240 + 0.25 Reorder the terms: 0.25 + 2x + x2 = 240 + 0.25 Combine like terms: 240 + 0.25 = 240.25 0.25 + 2x + x2 = 240.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 240.25 Calculate the square root of the right side: 15.5 Break this problem into two subproblems by setting (x + 0.5) equal to 15.5 and -15.5.Subproblem 1
x + 0.5 = 15.5 Simplifying x + 0.5 = 15.5 Reorder the terms: 0.5 + x = 15.5 Solving 0.5 + x = 15.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 15.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 15.5 + -0.5 x = 15.5 + -0.5 Combine like terms: 15.5 + -0.5 = 15 x = 15 Simplifying x = 15Subproblem 2
x + 0.5 = -15.5 Simplifying x + 0.5 = -15.5 Reorder the terms: 0.5 + x = -15.5 Solving 0.5 + x = -15.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -15.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -15.5 + -0.5 x = -15.5 + -0.5 Combine like terms: -15.5 + -0.5 = -16 x = -16 Simplifying x = -16Solution
The solution to the problem is based on the solutions from the subproblems. x = {15, -16}
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