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Simplifying x + x(2x + 1) = 31 Reorder the terms: x + x(1 + 2x) = 31 x + (1 * x + 2x * x) = 31 x + (1x + 2x2) = 31 Combine like terms: x + 1x = 2x 2x + 2x2 = 31 Solving 2x + 2x2 = 31 Solving for variable 'x'. Reorder the terms: -31 + 2x + 2x2 = 31 + -31 Combine like terms: 31 + -31 = 0 -31 + 2x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -15.5 + x + x2 = 0 Move the constant term to the right: Add '15.5' to each side of the equation. -15.5 + x + 15.5 + x2 = 0 + 15.5 Reorder the terms: -15.5 + 15.5 + x + x2 = 0 + 15.5 Combine like terms: -15.5 + 15.5 = 0.0 0.0 + x + x2 = 0 + 15.5 x + x2 = 0 + 15.5 Combine like terms: 0 + 15.5 = 15.5 x + x2 = 15.5 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. + 0.25 + x2 = 15.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 15.5 + 0.25 Combine like terms: 15.5 + 0.25 = 15.75 1.25 + x2 = 15.75 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 15.75 Calculate the square root of the right side: 3.968626967 Break this problem into two subproblems by setting (x + 0.5) equal to 3.968626967 and -3.968626967.Subproblem 1
x + 0.5 = 3.968626967 Simplifying x + 0.5 = 3.968626967 Reorder the terms: 0.5 + x = 3.968626967 Solving 0.5 + x = 3.968626967 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 = 3.968626967 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.968626967 + -0.5 x = 3.968626967 + -0.5 Combine like terms: 3.968626967 + -0.5 = 3.468626967 x = 3.468626967 Simplifying x = 3.468626967Subproblem 2
x + 0.5 = -3.968626967 Simplifying x + 0.5 = -3.968626967 Reorder the terms: 0.5 + x = -3.968626967 Solving 0.5 + x = -3.968626967 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 = -3.968626967 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.968626967 + -0.5 x = -3.968626967 + -0.5 Combine like terms: -3.968626967 + -0.5 = -4.468626967 x = -4.468626967 Simplifying x = -4.468626967Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.468626967, -4.468626967}
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