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Simplifying 0.99x2 + -2x + 1 = 0 Reorder the terms: 1 + -2x + 0.99x2 = 0 Solving 1 + -2x + 0.99x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.99 the coefficient of the squared term: Divide each side by '0.99'. 1.01010101 + -2.02020202x + x2 = 0 Move the constant term to the right: Add '-1.01010101' to each side of the equation. 1.01010101 + -2.02020202x + -1.01010101 + x2 = 0 + -1.01010101 Reorder the terms: 1.01010101 + -1.01010101 + -2.02020202x + x2 = 0 + -1.01010101 Combine like terms: 1.01010101 + -1.01010101 = 0.00000000 0.00000000 + -2.02020202x + x2 = 0 + -1.01010101 -2.02020202x + x2 = 0 + -1.01010101 Combine like terms: 0 + -1.01010101 = -1.01010101 -2.02020202x + x2 = -1.01010101 The x term is -2.02020202x. Take half its coefficient (-1.01010101). Square it (1.020304050) and add it to both sides. Add '1.020304050' to each side of the equation. -2.02020202x + 1.020304050 + x2 = -1.01010101 + 1.020304050 Reorder the terms: 1.020304050 + -2.02020202x + x2 = -1.01010101 + 1.020304050 Combine like terms: -1.01010101 + 1.020304050 = 0.01020304 1.020304050 + -2.02020202x + x2 = 0.01020304 Factor a perfect square on the left side: (x + -1.01010101)(x + -1.01010101) = 0.01020304 Calculate the square root of the right side: 0.101010099 Break this problem into two subproblems by setting (x + -1.01010101) equal to 0.101010099 and -0.101010099.Subproblem 1
x + -1.01010101 = 0.101010099 Simplifying x + -1.01010101 = 0.101010099 Reorder the terms: -1.01010101 + x = 0.101010099 Solving -1.01010101 + x = 0.101010099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.01010101' to each side of the equation. -1.01010101 + 1.01010101 + x = 0.101010099 + 1.01010101 Combine like terms: -1.01010101 + 1.01010101 = 0.00000000 0.00000000 + x = 0.101010099 + 1.01010101 x = 0.101010099 + 1.01010101 Combine like terms: 0.101010099 + 1.01010101 = 1.111111109 x = 1.111111109 Simplifying x = 1.111111109Subproblem 2
x + -1.01010101 = -0.101010099 Simplifying x + -1.01010101 = -0.101010099 Reorder the terms: -1.01010101 + x = -0.101010099 Solving -1.01010101 + x = -0.101010099 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.01010101' to each side of the equation. -1.01010101 + 1.01010101 + x = -0.101010099 + 1.01010101 Combine like terms: -1.01010101 + 1.01010101 = 0.00000000 0.00000000 + x = -0.101010099 + 1.01010101 x = -0.101010099 + 1.01010101 Combine like terms: -0.101010099 + 1.01010101 = 0.909090911 x = 0.909090911 Simplifying x = 0.909090911Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.111111109, 0.909090911}
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