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Simplifying (0.5x + 1)(x + 1) = 10 Reorder the terms: (1 + 0.5x)(x + 1) = 10 Reorder the terms: (1 + 0.5x)(1 + x) = 10 Multiply (1 + 0.5x) * (1 + x) (1(1 + x) + 0.5x * (1 + x)) = 10 ((1 * 1 + x * 1) + 0.5x * (1 + x)) = 10 ((1 + 1x) + 0.5x * (1 + x)) = 10 (1 + 1x + (1 * 0.5x + x * 0.5x)) = 10 (1 + 1x + (0.5x + 0.5x2)) = 10 Combine like terms: 1x + 0.5x = 1.5x (1 + 1.5x + 0.5x2) = 10 Solving 1 + 1.5x + 0.5x2 = 10 Solving for variable 'x'. Reorder the terms: 1 + -10 + 1.5x + 0.5x2 = 10 + -10 Combine like terms: 1 + -10 = -9 -9 + 1.5x + 0.5x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -9 + 1.5x + 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'. -18 + 3x + x2 = 0 Move the constant term to the right: Add '18' to each side of the equation. -18 + 3x + 18 + x2 = 0 + 18 Reorder the terms: -18 + 18 + 3x + x2 = 0 + 18 Combine like terms: -18 + 18 = 0 0 + 3x + x2 = 0 + 18 3x + x2 = 0 + 18 Combine like terms: 0 + 18 = 18 3x + x2 = 18 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 18 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 18 + 2.25 Combine like terms: 18 + 2.25 = 20.25 2.25 + 3x + x2 = 20.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 20.25 Calculate the square root of the right side: 4.5 Break this problem into two subproblems by setting (x + 1.5) equal to 4.5 and -4.5.Subproblem 1
x + 1.5 = 4.5 Simplifying x + 1.5 = 4.5 Reorder the terms: 1.5 + x = 4.5 Solving 1.5 + x = 4.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 4.5 + -1.5 x = 4.5 + -1.5 Combine like terms: 4.5 + -1.5 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 1.5 = -4.5 Simplifying x + 1.5 = -4.5 Reorder the terms: 1.5 + x = -4.5 Solving 1.5 + x = -4.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -4.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -4.5 + -1.5 x = -4.5 + -1.5 Combine like terms: -4.5 + -1.5 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -6}
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