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Simplifying 22x2 + x = 24 Reorder the terms: x + 22x2 = 24 Solving x + 22x2 = 24 Solving for variable 'x'. Reorder the terms: -24 + x + 22x2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + x + 22x2 = 0 Begin completing the square. Divide all terms by 22 the coefficient of the squared term: Divide each side by '22'. -1.090909091 + 0.04545454545x + x2 = 0 Move the constant term to the right: Add '1.090909091' to each side of the equation. -1.090909091 + 0.04545454545x + 1.090909091 + x2 = 0 + 1.090909091 Reorder the terms: -1.090909091 + 1.090909091 + 0.04545454545x + x2 = 0 + 1.090909091 Combine like terms: -1.090909091 + 1.090909091 = 0.000000000 0.000000000 + 0.04545454545x + x2 = 0 + 1.090909091 0.04545454545x + x2 = 0 + 1.090909091 Combine like terms: 0 + 1.090909091 = 1.090909091 0.04545454545x + x2 = 1.090909091 The x term is x. 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. 0.04545454545x + 0.25 + x2 = 1.090909091 + 0.25 Reorder the terms: 0.25 + 0.04545454545x + x2 = 1.090909091 + 0.25 Combine like terms: 1.090909091 + 0.25 = 1.340909091 0.25 + 0.04545454545x + x2 = 1.340909091 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 1.340909091 Calculate the square root of the right side: 1.157976291 Break this problem into two subproblems by setting (x + 0.5) equal to 1.157976291 and -1.157976291.Subproblem 1
x + 0.5 = 1.157976291 Simplifying x + 0.5 = 1.157976291 Reorder the terms: 0.5 + x = 1.157976291 Solving 0.5 + x = 1.157976291 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 = 1.157976291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.157976291 + -0.5 x = 1.157976291 + -0.5 Combine like terms: 1.157976291 + -0.5 = 0.657976291 x = 0.657976291 Simplifying x = 0.657976291Subproblem 2
x + 0.5 = -1.157976291 Simplifying x + 0.5 = -1.157976291 Reorder the terms: 0.5 + x = -1.157976291 Solving 0.5 + x = -1.157976291 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 = -1.157976291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.157976291 + -0.5 x = -1.157976291 + -0.5 Combine like terms: -1.157976291 + -0.5 = -1.657976291 x = -1.657976291 Simplifying x = -1.657976291Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.657976291, -1.657976291}
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