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Simplifying 0 = x2 + -1.998x + 0.998 Reorder the terms: 0 = 0.998 + -1.998x + x2 Solving 0 = 0.998 + -1.998x + x2 Solving for variable 'x'. Combine like terms: 0 + -0.998 = -0.998 -0.998 + 1.998x + -1x2 = 0.998 + -1.998x + x2 + -0.998 + 1.998x + -1x2 Reorder the terms: -0.998 + 1.998x + -1x2 = 0.998 + -0.998 + -1.998x + 1.998x + x2 + -1x2 Combine like terms: 0.998 + -0.998 = 0.000 -0.998 + 1.998x + -1x2 = 0.000 + -1.998x + 1.998x + x2 + -1x2 -0.998 + 1.998x + -1x2 = -1.998x + 1.998x + x2 + -1x2 Combine like terms: -1.998x + 1.998x = 0.000 -0.998 + 1.998x + -1x2 = 0.000 + x2 + -1x2 -0.998 + 1.998x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -0.998 + 1.998x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.998 + -1.998x + x2 = 0 Move the constant term to the right: Add '-0.998' to each side of the equation. 0.998 + -1.998x + -0.998 + x2 = 0 + -0.998 Reorder the terms: 0.998 + -0.998 + -1.998x + x2 = 0 + -0.998 Combine like terms: 0.998 + -0.998 = 0.000 0.000 + -1.998x + x2 = 0 + -0.998 -1.998x + x2 = 0 + -0.998 Combine like terms: 0 + -0.998 = -0.998 -1.998x + x2 = -0.998 The x term is -1.998x. Take half its coefficient (-0.999). Square it (0.998001) and add it to both sides. Add '0.998001' to each side of the equation. -1.998x + 0.998001 + x2 = -0.998 + 0.998001 Reorder the terms: 0.998001 + -1.998x + x2 = -0.998 + 0.998001 Combine like terms: -0.998 + 0.998001 = 0.000001 0.998001 + -1.998x + x2 = 0.000001 Factor a perfect square on the left side: (x + -0.999)(x + -0.999) = 0.000001 Calculate the square root of the right side: 0.001 Break this problem into two subproblems by setting (x + -0.999) equal to 0.001 and -0.001.Subproblem 1
x + -0.999 = 0.001 Simplifying x + -0.999 = 0.001 Reorder the terms: -0.999 + x = 0.001 Solving -0.999 + x = 0.001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.999' to each side of the equation. -0.999 + 0.999 + x = 0.001 + 0.999 Combine like terms: -0.999 + 0.999 = 0.000 0.000 + x = 0.001 + 0.999 x = 0.001 + 0.999 Combine like terms: 0.001 + 0.999 = 1 x = 1 Simplifying x = 1Subproblem 2
x + -0.999 = -0.001 Simplifying x + -0.999 = -0.001 Reorder the terms: -0.999 + x = -0.001 Solving -0.999 + x = -0.001 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.999' to each side of the equation. -0.999 + 0.999 + x = -0.001 + 0.999 Combine like terms: -0.999 + 0.999 = 0.000 0.000 + x = -0.001 + 0.999 x = -0.001 + 0.999 Combine like terms: -0.001 + 0.999 = 0.998 x = 0.998 Simplifying x = 0.998Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, 0.998}
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