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Simplifying x2 + x + 0.25 = 75 Reorder the terms: 0.25 + x + x2 = 75 Solving 0.25 + x + x2 = 75 Solving for variable 'x'. Reorder the terms: 0.25 + -75 + x + x2 = 75 + -75 Combine like terms: 0.25 + -75 = -74.75 -74.75 + x + x2 = 75 + -75 Combine like terms: 75 + -75 = 0 -74.75 + x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '74.75' to each side of the equation. -74.75 + x + 74.75 + x2 = 0 + 74.75 Reorder the terms: -74.75 + 74.75 + x + x2 = 0 + 74.75 Combine like terms: -74.75 + 74.75 = 0.00 0.00 + x + x2 = 0 + 74.75 x + x2 = 0 + 74.75 Combine like terms: 0 + 74.75 = 74.75 x + x2 = 74.75 The x term is x. 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 = 74.75 + 0.25 Reorder the terms: 0.25 + x + x2 = 74.75 + 0.25 Combine like terms: 74.75 + 0.25 = 75 0.25 + x + x2 = 75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 75 Calculate the square root of the right side: 8.660254038 Break this problem into two subproblems by setting (x + 0.5) equal to 8.660254038 and -8.660254038.Subproblem 1
x + 0.5 = 8.660254038 Simplifying x + 0.5 = 8.660254038 Reorder the terms: 0.5 + x = 8.660254038 Solving 0.5 + x = 8.660254038 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 = 8.660254038 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 8.660254038 + -0.5 x = 8.660254038 + -0.5 Combine like terms: 8.660254038 + -0.5 = 8.160254038 x = 8.160254038 Simplifying x = 8.160254038Subproblem 2
x + 0.5 = -8.660254038 Simplifying x + 0.5 = -8.660254038 Reorder the terms: 0.5 + x = -8.660254038 Solving 0.5 + x = -8.660254038 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 = -8.660254038 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -8.660254038 + -0.5 x = -8.660254038 + -0.5 Combine like terms: -8.660254038 + -0.5 = -9.160254038 x = -9.160254038 Simplifying x = -9.160254038Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.160254038, -9.160254038}
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