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Simplifying x2 + 75 = x + 625 Reorder the terms: 75 + x2 = x + 625 Reorder the terms: 75 + x2 = 625 + x Solving 75 + x2 = 625 + x Solving for variable 'x'. Reorder the terms: 75 + -625 + -1x + x2 = 625 + x + -625 + -1x Combine like terms: 75 + -625 = -550 -550 + -1x + x2 = 625 + x + -625 + -1x Reorder the terms: -550 + -1x + x2 = 625 + -625 + x + -1x Combine like terms: 625 + -625 = 0 -550 + -1x + x2 = 0 + x + -1x -550 + -1x + x2 = x + -1x Combine like terms: x + -1x = 0 -550 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '550' to each side of the equation. -550 + -1x + 550 + x2 = 0 + 550 Reorder the terms: -550 + 550 + -1x + x2 = 0 + 550 Combine like terms: -550 + 550 = 0 0 + -1x + x2 = 0 + 550 -1x + x2 = 0 + 550 Combine like terms: 0 + 550 = 550 -1x + x2 = 550 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. -1x + 0.25 + x2 = 550 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 550 + 0.25 Combine like terms: 550 + 0.25 = 550.25 0.25 + -1x + x2 = 550.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 550.25 Calculate the square root of the right side: 23.457408211 Break this problem into two subproblems by setting (x + -0.5) equal to 23.457408211 and -23.457408211.Subproblem 1
x + -0.5 = 23.457408211 Simplifying x + -0.5 = 23.457408211 Reorder the terms: -0.5 + x = 23.457408211 Solving -0.5 + x = 23.457408211 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 = 23.457408211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 23.457408211 + 0.5 x = 23.457408211 + 0.5 Combine like terms: 23.457408211 + 0.5 = 23.957408211 x = 23.957408211 Simplifying x = 23.957408211Subproblem 2
x + -0.5 = -23.457408211 Simplifying x + -0.5 = -23.457408211 Reorder the terms: -0.5 + x = -23.457408211 Solving -0.5 + x = -23.457408211 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 = -23.457408211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -23.457408211 + 0.5 x = -23.457408211 + 0.5 Combine like terms: -23.457408211 + 0.5 = -22.957408211 x = -22.957408211 Simplifying x = -22.957408211Solution
The solution to the problem is based on the solutions from the subproblems. x = {23.957408211, -22.957408211}
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