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Simplifying x(x + 1) = 75 Reorder the terms: x(1 + x) = 75 (1 * x + x * x) = 75 (1x + x2) = 75 Solving 1x + x2 = 75 Solving for variable 'x'. Reorder the terms: -75 + 1x + x2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '75' to each side of the equation. -75 + 1x + 75 + x2 = 0 + 75 Reorder the terms: -75 + 75 + 1x + x2 = 0 + 75 Combine like terms: -75 + 75 = 0 0 + 1x + x2 = 0 + 75 1x + x2 = 0 + 75 Combine like terms: 0 + 75 = 75 1x + x2 = 75 The x term is 1x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. x + 0.25 + x2 = 75 + 0.25 Reorder the terms: 0.25 + x + x2 = 75 + 0.25 Combine like terms: 75 + 0.25 = 75.25 0.25 + x + x2 = 75.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 75.25 Calculate the square root of the right side: 8.674675786 Break this problem into two subproblems by setting (x + 0.5) equal to 8.674675786 and -8.674675786.Subproblem 1
x + 0.5 = 8.674675786 Simplifying x + 0.5 = 8.674675786 Reorder the terms: 0.5 + x = 8.674675786 Solving 0.5 + x = 8.674675786 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.674675786 + -0.5 x = 8.674675786 + -0.5 Combine like terms: 8.674675786 + -0.5 = 8.174675786 x = 8.174675786 Simplifying x = 8.174675786Subproblem 2
x + 0.5 = -8.674675786 Simplifying x + 0.5 = -8.674675786 Reorder the terms: 0.5 + x = -8.674675786 Solving 0.5 + x = -8.674675786 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -8.674675786 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.674675786 + -0.5 x = -8.674675786 + -0.5 Combine like terms: -8.674675786 + -0.5 = -9.174675786 x = -9.174675786 Simplifying x = -9.174675786Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.174675786, -9.174675786}
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