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Simplifying (10 + 0.5x) * x = 48 Reorder the terms for easier multiplication: x(10 + 0.5x) = 48 (10 * x + 0.5x * x) = 48 (10x + 0.5x2) = 48 Solving 10x + 0.5x2 = 48 Solving for variable 'x'. Reorder the terms: -48 + 10x + 0.5x2 = 48 + -48 Combine like terms: 48 + -48 = 0 -48 + 10x + 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'. -96 + 20x + x2 = 0 Move the constant term to the right: Add '96' to each side of the equation. -96 + 20x + 96 + x2 = 0 + 96 Reorder the terms: -96 + 96 + 20x + x2 = 0 + 96 Combine like terms: -96 + 96 = 0 0 + 20x + x2 = 0 + 96 20x + x2 = 0 + 96 Combine like terms: 0 + 96 = 96 20x + x2 = 96 The x term is 20x. Take half its coefficient (10). Square it (100) and add it to both sides. Add '100' to each side of the equation. 20x + 100 + x2 = 96 + 100 Reorder the terms: 100 + 20x + x2 = 96 + 100 Combine like terms: 96 + 100 = 196 100 + 20x + x2 = 196 Factor a perfect square on the left side: (x + 10)(x + 10) = 196 Calculate the square root of the right side: 14 Break this problem into two subproblems by setting (x + 10) equal to 14 and -14.Subproblem 1
x + 10 = 14 Simplifying x + 10 = 14 Reorder the terms: 10 + x = 14 Solving 10 + x = 14 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = 14 + -10 Combine like terms: 10 + -10 = 0 0 + x = 14 + -10 x = 14 + -10 Combine like terms: 14 + -10 = 4 x = 4 Simplifying x = 4Subproblem 2
x + 10 = -14 Simplifying x + 10 = -14 Reorder the terms: 10 + x = -14 Solving 10 + x = -14 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + x = -14 + -10 Combine like terms: 10 + -10 = 0 0 + x = -14 + -10 x = -14 + -10 Combine like terms: -14 + -10 = -24 x = -24 Simplifying x = -24Solution
The solution to the problem is based on the solutions from the subproblems. x = {4, -24}
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