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Simplifying x2 + x = 412 Reorder the terms: x + x2 = 412 Solving x + x2 = 412 Solving for variable 'x'. Reorder the terms: -412 + x + x2 = 412 + -412 Combine like terms: 412 + -412 = 0 -412 + x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '412' to each side of the equation. -412 + x + 412 + x2 = 0 + 412 Reorder the terms: -412 + 412 + x + x2 = 0 + 412 Combine like terms: -412 + 412 = 0 0 + x + x2 = 0 + 412 x + x2 = 0 + 412 Combine like terms: 0 + 412 = 412 x + x2 = 412 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 = 412 + 0.25 Reorder the terms: 0.25 + x + x2 = 412 + 0.25 Combine like terms: 412 + 0.25 = 412.25 0.25 + x + x2 = 412.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 412.25 Calculate the square root of the right side: 20.303940504 Break this problem into two subproblems by setting (x + 0.5) equal to 20.303940504 and -20.303940504.Subproblem 1
x + 0.5 = 20.303940504 Simplifying x + 0.5 = 20.303940504 Reorder the terms: 0.5 + x = 20.303940504 Solving 0.5 + x = 20.303940504 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 = 20.303940504 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 20.303940504 + -0.5 x = 20.303940504 + -0.5 Combine like terms: 20.303940504 + -0.5 = 19.803940504 x = 19.803940504 Simplifying x = 19.803940504Subproblem 2
x + 0.5 = -20.303940504 Simplifying x + 0.5 = -20.303940504 Reorder the terms: 0.5 + x = -20.303940504 Solving 0.5 + x = -20.303940504 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 = -20.303940504 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -20.303940504 + -0.5 x = -20.303940504 + -0.5 Combine like terms: -20.303940504 + -0.5 = -20.803940504 x = -20.803940504 Simplifying x = -20.803940504Solution
The solution to the problem is based on the solutions from the subproblems. x = {19.803940504, -20.803940504}
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