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Simplifying 2x + 30 = -0.05x2 + 1.5x + 48.75 Reorder the terms: 30 + 2x = -0.05x2 + 1.5x + 48.75 Reorder the terms: 30 + 2x = 48.75 + 1.5x + -0.05x2 Solving 30 + 2x = 48.75 + 1.5x + -0.05x2 Solving for variable 'x'. Reorder the terms: 30 + -48.75 + 2x + -1.5x + 0.05x2 = 48.75 + 1.5x + -0.05x2 + -48.75 + -1.5x + 0.05x2 Combine like terms: 30 + -48.75 = -18.75 -18.75 + 2x + -1.5x + 0.05x2 = 48.75 + 1.5x + -0.05x2 + -48.75 + -1.5x + 0.05x2 Combine like terms: 2x + -1.5x = 0.5x -18.75 + 0.5x + 0.05x2 = 48.75 + 1.5x + -0.05x2 + -48.75 + -1.5x + 0.05x2 Reorder the terms: -18.75 + 0.5x + 0.05x2 = 48.75 + -48.75 + 1.5x + -1.5x + -0.05x2 + 0.05x2 Combine like terms: 48.75 + -48.75 = 0.00 -18.75 + 0.5x + 0.05x2 = 0.00 + 1.5x + -1.5x + -0.05x2 + 0.05x2 -18.75 + 0.5x + 0.05x2 = 1.5x + -1.5x + -0.05x2 + 0.05x2 Combine like terms: 1.5x + -1.5x = 0.0 -18.75 + 0.5x + 0.05x2 = 0.0 + -0.05x2 + 0.05x2 -18.75 + 0.5x + 0.05x2 = -0.05x2 + 0.05x2 Combine like terms: -0.05x2 + 0.05x2 = 0.00 -18.75 + 0.5x + 0.05x2 = 0.00 Begin completing the square. Divide all terms by 0.05 the coefficient of the squared term: Divide each side by '0.05'. -375 + 10x + x2 = 0 Move the constant term to the right: Add '375' to each side of the equation. -375 + 10x + 375 + x2 = 0 + 375 Reorder the terms: -375 + 375 + 10x + x2 = 0 + 375 Combine like terms: -375 + 375 = 0 0 + 10x + x2 = 0 + 375 10x + x2 = 0 + 375 Combine like terms: 0 + 375 = 375 10x + x2 = 375 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 = 375 + 25 Reorder the terms: 25 + 10x + x2 = 375 + 25 Combine like terms: 375 + 25 = 400 25 + 10x + x2 = 400 Factor a perfect square on the left side: (x + 5)(x + 5) = 400 Calculate the square root of the right side: 20 Break this problem into two subproblems by setting (x + 5) equal to 20 and -20.Subproblem 1
x + 5 = 20 Simplifying x + 5 = 20 Reorder the terms: 5 + x = 20 Solving 5 + x = 20 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 = 20 + -5 Combine like terms: 5 + -5 = 0 0 + x = 20 + -5 x = 20 + -5 Combine like terms: 20 + -5 = 15 x = 15 Simplifying x = 15Subproblem 2
x + 5 = -20 Simplifying x + 5 = -20 Reorder the terms: 5 + x = -20 Solving 5 + x = -20 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 = -20 + -5 Combine like terms: 5 + -5 = 0 0 + x = -20 + -5 x = -20 + -5 Combine like terms: -20 + -5 = -25 x = -25 Simplifying x = -25Solution
The solution to the problem is based on the solutions from the subproblems. x = {15, -25}
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