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Simplifying 0.5x(x + 1) = 36 Reorder the terms: 0.5x(1 + x) = 36 (1 * 0.5x + x * 0.5x) = 36 (0.5x + 0.5x2) = 36 Solving 0.5x + 0.5x2 = 36 Solving for variable 'x'. Reorder the terms: -36 + 0.5x + 0.5x2 = 36 + -36 Combine like terms: 36 + -36 = 0 -36 + 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'. -72 + x + x2 = 0 Move the constant term to the right: Add '72' to each side of the equation. -72 + x + 72 + x2 = 0 + 72 Reorder the terms: -72 + 72 + x + x2 = 0 + 72 Combine like terms: -72 + 72 = 0 0 + x + x2 = 0 + 72 x + x2 = 0 + 72 Combine like terms: 0 + 72 = 72 x + x2 = 72 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 = 72 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 72 + 0.25 Combine like terms: 72 + 0.25 = 72.25 1.25 + x2 = 72.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + 0.5) equal to 8.5 and -8.5.Subproblem 1
x + 0.5 = 8.5 Simplifying x + 0.5 = 8.5 Reorder the terms: 0.5 + x = 8.5 Solving 0.5 + x = 8.5 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 = 8.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.5 + -0.5 x = 8.5 + -0.5 Combine like terms: 8.5 + -0.5 = 8 x = 8 Simplifying x = 8Subproblem 2
x + 0.5 = -8.5 Simplifying x + 0.5 = -8.5 Reorder the terms: 0.5 + x = -8.5 Solving 0.5 + x = -8.5 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 = -8.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.5 + -0.5 x = -8.5 + -0.5 Combine like terms: -8.5 + -0.5 = -9 x = -9 Simplifying x = -9Solution
The solution to the problem is based on the solutions from the subproblems. x = {8, -9}
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