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Simplifying 756 = 450 + 0.5x(x + 1) Reorder the terms: 756 = 450 + 0.5x(1 + x) 756 = 450 + (1 * 0.5x + x * 0.5x) 756 = 450 + (0.5x + 0.5x2) Solving 756 = 450 + 0.5x + 0.5x2 Solving for variable 'x'. Combine like terms: 756 + -450 = 306 306 + -0.5x + -0.5x2 = 450 + 0.5x + 0.5x2 + -450 + -0.5x + -0.5x2 Reorder the terms: 306 + -0.5x + -0.5x2 = 450 + -450 + 0.5x + -0.5x + 0.5x2 + -0.5x2 Combine like terms: 450 + -450 = 0 306 + -0.5x + -0.5x2 = 0 + 0.5x + -0.5x + 0.5x2 + -0.5x2 306 + -0.5x + -0.5x2 = 0.5x + -0.5x + 0.5x2 + -0.5x2 Combine like terms: 0.5x + -0.5x = 0.0 306 + -0.5x + -0.5x2 = 0.0 + 0.5x2 + -0.5x2 306 + -0.5x + -0.5x2 = 0.5x2 + -0.5x2 Combine like terms: 0.5x2 + -0.5x2 = 0.0 306 + -0.5x + -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'. -612 + x + x2 = 0 Move the constant term to the right: Add '612' to each side of the equation. -612 + x + 612 + x2 = 0 + 612 Reorder the terms: -612 + 612 + x + x2 = 0 + 612 Combine like terms: -612 + 612 = 0 0 + x + x2 = 0 + 612 x + x2 = 0 + 612 Combine like terms: 0 + 612 = 612 x + x2 = 612 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 = 612 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 612 + 0.25 Combine like terms: 612 + 0.25 = 612.25 1.25 + x2 = 612.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 612.25 Calculate the square root of the right side: 24.743686063 Break this problem into two subproblems by setting (x + 0.5) equal to 24.743686063 and -24.743686063.Subproblem 1
x + 0.5 = 24.743686063 Simplifying x + 0.5 = 24.743686063 Reorder the terms: 0.5 + x = 24.743686063 Solving 0.5 + x = 24.743686063 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 = 24.743686063 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 24.743686063 + -0.5 x = 24.743686063 + -0.5 Combine like terms: 24.743686063 + -0.5 = 24.243686063 x = 24.243686063 Simplifying x = 24.243686063Subproblem 2
x + 0.5 = -24.743686063 Simplifying x + 0.5 = -24.743686063 Reorder the terms: 0.5 + x = -24.743686063 Solving 0.5 + x = -24.743686063 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 = -24.743686063 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -24.743686063 + -0.5 x = -24.743686063 + -0.5 Combine like terms: -24.743686063 + -0.5 = -25.243686063 x = -25.243686063 Simplifying x = -25.243686063Solution
The solution to the problem is based on the solutions from the subproblems. x = {24.243686063, -25.243686063}
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