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Simplifying x(2x + 1) = 68 Reorder the terms: x(1 + 2x) = 68 (1 * x + 2x * x) = 68 (1x + 2x2) = 68 Solving 1x + 2x2 = 68 Solving for variable 'x'. Reorder the terms: -68 + 1x + 2x2 = 68 + -68 Combine like terms: 68 + -68 = 0 -68 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -34 + 0.5x + x2 = 0 Move the constant term to the right: Add '34' to each side of the equation. -34 + 0.5x + 34 + x2 = 0 + 34 Reorder the terms: -34 + 34 + 0.5x + x2 = 0 + 34 Combine like terms: -34 + 34 = 0 0 + 0.5x + x2 = 0 + 34 0.5x + x2 = 0 + 34 Combine like terms: 0 + 34 = 34 0.5x + x2 = 34 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 34 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 34 + 0.0625 Combine like terms: 34 + 0.0625 = 34.0625 0.0625 + 0.5x + x2 = 34.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 34.0625 Calculate the square root of the right side: 5.836308765 Break this problem into two subproblems by setting (x + 0.25) equal to 5.836308765 and -5.836308765.Subproblem 1
x + 0.25 = 5.836308765 Simplifying x + 0.25 = 5.836308765 Reorder the terms: 0.25 + x = 5.836308765 Solving 0.25 + x = 5.836308765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.836308765 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.836308765 + -0.25 x = 5.836308765 + -0.25 Combine like terms: 5.836308765 + -0.25 = 5.586308765 x = 5.586308765 Simplifying x = 5.586308765Subproblem 2
x + 0.25 = -5.836308765 Simplifying x + 0.25 = -5.836308765 Reorder the terms: 0.25 + x = -5.836308765 Solving 0.25 + x = -5.836308765 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.836308765 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.836308765 + -0.25 x = -5.836308765 + -0.25 Combine like terms: -5.836308765 + -0.25 = -6.086308765 x = -6.086308765 Simplifying x = -6.086308765Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.586308765, -6.086308765}
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