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Simplifying 0.5(x(x + 20)) = 78 Reorder the terms: 0.5(x(20 + x)) = 78 0.5((20 * x + x * x)) = 78 0.5((20x + x2)) = 78 (20x * 0.5 + x2 * 0.5) = 78 (10x + 0.5x2) = 78 Solving 10x + 0.5x2 = 78 Solving for variable 'x'. Reorder the terms: -78 + 10x + 0.5x2 = 78 + -78 Combine like terms: 78 + -78 = 0 -78 + 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'. -156 + 20x + x2 = 0 Move the constant term to the right: Add '156' to each side of the equation. -156 + 20x + 156 + x2 = 0 + 156 Reorder the terms: -156 + 156 + 20x + x2 = 0 + 156 Combine like terms: -156 + 156 = 0 0 + 20x + x2 = 0 + 156 20x + x2 = 0 + 156 Combine like terms: 0 + 156 = 156 20x + x2 = 156 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 = 156 + 100 Reorder the terms: 100 + 20x + x2 = 156 + 100 Combine like terms: 156 + 100 = 256 100 + 20x + x2 = 256 Factor a perfect square on the left side: (x + 10)(x + 10) = 256 Calculate the square root of the right side: 16 Break this problem into two subproblems by setting (x + 10) equal to 16 and -16.Subproblem 1
x + 10 = 16 Simplifying x + 10 = 16 Reorder the terms: 10 + x = 16 Solving 10 + x = 16 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 = 16 + -10 Combine like terms: 10 + -10 = 0 0 + x = 16 + -10 x = 16 + -10 Combine like terms: 16 + -10 = 6 x = 6 Simplifying x = 6Subproblem 2
x + 10 = -16 Simplifying x + 10 = -16 Reorder the terms: 10 + x = -16 Solving 10 + x = -16 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 = -16 + -10 Combine like terms: 10 + -10 = 0 0 + x = -16 + -10 x = -16 + -10 Combine like terms: -16 + -10 = -26 x = -26 Simplifying x = -26Solution
The solution to the problem is based on the solutions from the subproblems. x = {6, -26}
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