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Simplifying 10x2 + -41x + 32 = 0 Reorder the terms: 32 + -41x + 10x2 = 0 Solving 32 + -41x + 10x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 3.2 + -4.1x + x2 = 0 Move the constant term to the right: Add '-3.2' to each side of the equation. 3.2 + -4.1x + -3.2 + x2 = 0 + -3.2 Reorder the terms: 3.2 + -3.2 + -4.1x + x2 = 0 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + -4.1x + x2 = 0 + -3.2 -4.1x + x2 = 0 + -3.2 Combine like terms: 0 + -3.2 = -3.2 -4.1x + x2 = -3.2 The x term is -4.1x. Take half its coefficient (-2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. -4.1x + 4.2025 + x2 = -3.2 + 4.2025 Reorder the terms: 4.2025 + -4.1x + x2 = -3.2 + 4.2025 Combine like terms: -3.2 + 4.2025 = 1.0025 4.2025 + -4.1x + x2 = 1.0025 Factor a perfect square on the left side: (x + -2.05)(x + -2.05) = 1.0025 Calculate the square root of the right side: 1.00124922 Break this problem into two subproblems by setting (x + -2.05) equal to 1.00124922 and -1.00124922.Subproblem 1
x + -2.05 = 1.00124922 Simplifying x + -2.05 = 1.00124922 Reorder the terms: -2.05 + x = 1.00124922 Solving -2.05 + x = 1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + x = 1.00124922 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + x = 1.00124922 + 2.05 x = 1.00124922 + 2.05 Combine like terms: 1.00124922 + 2.05 = 3.05124922 x = 3.05124922 Simplifying x = 3.05124922Subproblem 2
x + -2.05 = -1.00124922 Simplifying x + -2.05 = -1.00124922 Reorder the terms: -2.05 + x = -1.00124922 Solving -2.05 + x = -1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + x = -1.00124922 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + x = -1.00124922 + 2.05 x = -1.00124922 + 2.05 Combine like terms: -1.00124922 + 2.05 = 1.04875078 x = 1.04875078 Simplifying x = 1.04875078Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.05124922, 1.04875078}
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