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Simplifying 4.11x2 + -9.22x + 4.11 = 0 Reorder the terms: 4.11 + -9.22x + 4.11x2 = 0 Solving 4.11 + -9.22x + 4.11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4.11 the coefficient of the squared term: Divide each side by '4.11'. 1 + -2.243309002x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + -2.243309002x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + -2.243309002x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + -2.243309002x + x2 = 0 + -1 -2.243309002x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 -2.243309002x + x2 = -1 The x term is -2.243309002x. Take half its coefficient (-1.121654501). Square it (1.258108820) and add it to both sides. Add '1.258108820' to each side of the equation. -2.243309002x + 1.258108820 + x2 = -1 + 1.258108820 Reorder the terms: 1.258108820 + -2.243309002x + x2 = -1 + 1.258108820 Combine like terms: -1 + 1.258108820 = 0.25810882 1.258108820 + -2.243309002x + x2 = 0.25810882 Factor a perfect square on the left side: (x + -1.121654501)(x + -1.121654501) = 0.25810882 Calculate the square root of the right side: 0.508044112 Break this problem into two subproblems by setting (x + -1.121654501) equal to 0.508044112 and -0.508044112.Subproblem 1
x + -1.121654501 = 0.508044112 Simplifying x + -1.121654501 = 0.508044112 Reorder the terms: -1.121654501 + x = 0.508044112 Solving -1.121654501 + x = 0.508044112 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.121654501' to each side of the equation. -1.121654501 + 1.121654501 + x = 0.508044112 + 1.121654501 Combine like terms: -1.121654501 + 1.121654501 = 0.000000000 0.000000000 + x = 0.508044112 + 1.121654501 x = 0.508044112 + 1.121654501 Combine like terms: 0.508044112 + 1.121654501 = 1.629698613 x = 1.629698613 Simplifying x = 1.629698613Subproblem 2
x + -1.121654501 = -0.508044112 Simplifying x + -1.121654501 = -0.508044112 Reorder the terms: -1.121654501 + x = -0.508044112 Solving -1.121654501 + x = -0.508044112 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.121654501' to each side of the equation. -1.121654501 + 1.121654501 + x = -0.508044112 + 1.121654501 Combine like terms: -1.121654501 + 1.121654501 = 0.000000000 0.000000000 + x = -0.508044112 + 1.121654501 x = -0.508044112 + 1.121654501 Combine like terms: -0.508044112 + 1.121654501 = 0.613610389 x = 0.613610389 Simplifying x = 0.613610389Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.629698613, 0.613610389}
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