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Simplifying 0 = 5.2x2 + -5.2x + 1 Reorder the terms: 0 = 1 + -5.2x + 5.2x2 Solving 0 = 1 + -5.2x + 5.2x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 5.2x + -5.2x2 = 1 + -5.2x + 5.2x2 + -1 + 5.2x + -5.2x2 Reorder the terms: -1 + 5.2x + -5.2x2 = 1 + -1 + -5.2x + 5.2x + 5.2x2 + -5.2x2 Combine like terms: 1 + -1 = 0 -1 + 5.2x + -5.2x2 = 0 + -5.2x + 5.2x + 5.2x2 + -5.2x2 -1 + 5.2x + -5.2x2 = -5.2x + 5.2x + 5.2x2 + -5.2x2 Combine like terms: -5.2x + 5.2x = 0.0 -1 + 5.2x + -5.2x2 = 0.0 + 5.2x2 + -5.2x2 -1 + 5.2x + -5.2x2 = 5.2x2 + -5.2x2 Combine like terms: 5.2x2 + -5.2x2 = 0.0 -1 + 5.2x + -5.2x2 = 0.0 Begin completing the square. Divide all terms by -5.2 the coefficient of the squared term: Divide each side by '-5.2'. 0.1923076923 + -1x + x2 = 0 Move the constant term to the right: Add '-0.1923076923' to each side of the equation. 0.1923076923 + -1x + -0.1923076923 + x2 = 0 + -0.1923076923 Reorder the terms: 0.1923076923 + -0.1923076923 + -1x + x2 = 0 + -0.1923076923 Combine like terms: 0.1923076923 + -0.1923076923 = 0.0000000000 0.0000000000 + -1x + x2 = 0 + -0.1923076923 -1x + x2 = 0 + -0.1923076923 Combine like terms: 0 + -0.1923076923 = -0.1923076923 -1x + x2 = -0.1923076923 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 = -0.1923076923 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.1923076923 + 0.25 Combine like terms: -0.1923076923 + 0.25 = 0.0576923077 0.25 + -1x + x2 = 0.0576923077 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.0576923077 Calculate the square root of the right side: 0.240192231 Break this problem into two subproblems by setting (x + -0.5) equal to 0.240192231 and -0.240192231.Subproblem 1
x + -0.5 = 0.240192231 Simplifying x + -0.5 = 0.240192231 Reorder the terms: -0.5 + x = 0.240192231 Solving -0.5 + x = 0.240192231 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 = 0.240192231 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.240192231 + 0.5 x = 0.240192231 + 0.5 Combine like terms: 0.240192231 + 0.5 = 0.740192231 x = 0.740192231 Simplifying x = 0.740192231Subproblem 2
x + -0.5 = -0.240192231 Simplifying x + -0.5 = -0.240192231 Reorder the terms: -0.5 + x = -0.240192231 Solving -0.5 + x = -0.240192231 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 = -0.240192231 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.240192231 + 0.5 x = -0.240192231 + 0.5 Combine like terms: -0.240192231 + 0.5 = 0.259807769 x = 0.259807769 Simplifying x = 0.259807769Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.740192231, 0.259807769}
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