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Simplifying x(x + 7.5) = 2(2x + 7.5) Reorder the terms: x(7.5 + x) = 2(2x + 7.5) (7.5 * x + x * x) = 2(2x + 7.5) (7.5x + x2) = 2(2x + 7.5) Reorder the terms: 7.5x + x2 = 2(7.5 + 2x) 7.5x + x2 = (7.5 * 2 + 2x * 2) 7.5x + x2 = (15 + 4x) Solving 7.5x + x2 = 15 + 4x Solving for variable 'x'. Reorder the terms: -15 + 7.5x + -4x + x2 = 15 + 4x + -15 + -4x Combine like terms: 7.5x + -4x = 3.5x -15 + 3.5x + x2 = 15 + 4x + -15 + -4x Reorder the terms: -15 + 3.5x + x2 = 15 + -15 + 4x + -4x Combine like terms: 15 + -15 = 0 -15 + 3.5x + x2 = 0 + 4x + -4x -15 + 3.5x + x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -15 + 3.5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '15' to each side of the equation. -15 + 3.5x + 15 + x2 = 0 + 15 Reorder the terms: -15 + 15 + 3.5x + x2 = 0 + 15 Combine like terms: -15 + 15 = 0 0 + 3.5x + x2 = 0 + 15 3.5x + x2 = 0 + 15 Combine like terms: 0 + 15 = 15 3.5x + x2 = 15 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 15 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 15 + 3.0625 Combine like terms: 15 + 3.0625 = 18.0625 3.0625 + 3.5x + x2 = 18.0625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 18.0625 Calculate the square root of the right side: 4.25 Break this problem into two subproblems by setting (x + 1.75) equal to 4.25 and -4.25.Subproblem 1
x + 1.75 = 4.25 Simplifying x + 1.75 = 4.25 Reorder the terms: 1.75 + x = 4.25 Solving 1.75 + x = 4.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 4.25 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 4.25 + -1.75 x = 4.25 + -1.75 Combine like terms: 4.25 + -1.75 = 2.5 x = 2.5 Simplifying x = 2.5Subproblem 2
x + 1.75 = -4.25 Simplifying x + 1.75 = -4.25 Reorder the terms: 1.75 + x = -4.25 Solving 1.75 + x = -4.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -4.25 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -4.25 + -1.75 x = -4.25 + -1.75 Combine like terms: -4.25 + -1.75 = -6 x = -6 Simplifying x = -6Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.5, -6}
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