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Simplifying 0.5x(x + 1) = 300 Reorder the terms: 0.5x(1 + x) = 300 (1 * 0.5x + x * 0.5x) = 300 (0.5x + 0.5x2) = 300 Solving 0.5x + 0.5x2 = 300 Solving for variable 'x'. Reorder the terms: -300 + 0.5x + 0.5x2 = 300 + -300 Combine like terms: 300 + -300 = 0 -300 + 0.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -600 + x + x2 = 0 Move the constant term to the right: Add '600' to each side of the equation. -600 + x + 600 + x2 = 0 + 600 Reorder the terms: -600 + 600 + x + x2 = 0 + 600 Combine like terms: -600 + 600 = 0 0 + x + x2 = 0 + 600 x + x2 = 0 + 600 Combine like terms: 0 + 600 = 600 x + x2 = 600 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. + 0.25 + x2 = 600 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 600 + 0.25 Combine like terms: 600 + 0.25 = 600.25 1.25 + x2 = 600.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 600.25 Calculate the square root of the right side: 24.5 Break this problem into two subproblems by setting (x + 0.5) equal to 24.5 and -24.5.Subproblem 1
x + 0.5 = 24.5 Simplifying x + 0.5 = 24.5 Reorder the terms: 0.5 + x = 24.5 Solving 0.5 + x = 24.5 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 = 24.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 24.5 + -0.5 x = 24.5 + -0.5 Combine like terms: 24.5 + -0.5 = 24 x = 24 Simplifying x = 24Subproblem 2
x + 0.5 = -24.5 Simplifying x + 0.5 = -24.5 Reorder the terms: 0.5 + x = -24.5 Solving 0.5 + x = -24.5 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 = -24.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -24.5 + -0.5 x = -24.5 + -0.5 Combine like terms: -24.5 + -0.5 = -25 x = -25 Simplifying x = -25Solution
The solution to the problem is based on the solutions from the subproblems. x = {24, -25}
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