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Simplifying 0.5(x + 5)(x + 5) = 18 Reorder the terms: 0.5(5 + x)(x + 5) = 18 Reorder the terms: 0.5(5 + x)(5 + x) = 18 Multiply (5 + x) * (5 + x) 0.5(5(5 + x) + x(5 + x)) = 18 0.5((5 * 5 + x * 5) + x(5 + x)) = 18 0.5((25 + 5x) + x(5 + x)) = 18 0.5(25 + 5x + (5 * x + x * x)) = 18 0.5(25 + 5x + (5x + x2)) = 18 Combine like terms: 5x + 5x = 10x 0.5(25 + 10x + x2) = 18 (25 * 0.5 + 10x * 0.5 + x2 * 0.5) = 18 (12.5 + 5x + 0.5x2) = 18 Solving 12.5 + 5x + 0.5x2 = 18 Solving for variable 'x'. Reorder the terms: 12.5 + -18 + 5x + 0.5x2 = 18 + -18 Combine like terms: 12.5 + -18 = -5.5 -5.5 + 5x + 0.5x2 = 18 + -18 Combine like terms: 18 + -18 = 0 -5.5 + 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'. -11 + 10x + x2 = 0 Move the constant term to the right: Add '11' to each side of the equation. -11 + 10x + 11 + x2 = 0 + 11 Reorder the terms: -11 + 11 + 10x + x2 = 0 + 11 Combine like terms: -11 + 11 = 0 0 + 10x + x2 = 0 + 11 10x + x2 = 0 + 11 Combine like terms: 0 + 11 = 11 10x + x2 = 11 The x term is 10x. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10x + 25 + x2 = 11 + 25 Reorder the terms: 25 + 10x + x2 = 11 + 25 Combine like terms: 11 + 25 = 36 25 + 10x + x2 = 36 Factor a perfect square on the left side: (x + 5)(x + 5) = 36 Calculate the square root of the right side: 6 Break this problem into two subproblems by setting (x + 5) equal to 6 and -6.Subproblem 1
x + 5 = 6 Simplifying x + 5 = 6 Reorder the terms: 5 + x = 6 Solving 5 + x = 6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = 6 + -5 Combine like terms: 5 + -5 = 0 0 + x = 6 + -5 x = 6 + -5 Combine like terms: 6 + -5 = 1 x = 1 Simplifying x = 1Subproblem 2
x + 5 = -6 Simplifying x + 5 = -6 Reorder the terms: 5 + x = -6 Solving 5 + x = -6 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + x = -6 + -5 Combine like terms: 5 + -5 = 0 0 + x = -6 + -5 x = -6 + -5 Combine like terms: -6 + -5 = -11 x = -11 Simplifying x = -11Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, -11}
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