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Simplifying 24 = (x + 1)(2x + 1) Reorder the terms: 24 = (1 + x)(2x + 1) Reorder the terms: 24 = (1 + x)(1 + 2x) Multiply (1 + x) * (1 + 2x) 24 = (1(1 + 2x) + x(1 + 2x)) 24 = ((1 * 1 + 2x * 1) + x(1 + 2x)) 24 = ((1 + 2x) + x(1 + 2x)) 24 = (1 + 2x + (1 * x + 2x * x)) 24 = (1 + 2x + (1x + 2x2)) Combine like terms: 2x + 1x = 3x 24 = (1 + 3x + 2x2) Solving 24 = 1 + 3x + 2x2 Solving for variable 'x'. Combine like terms: 24 + -1 = 23 23 + -3x + -2x2 = 1 + 3x + 2x2 + -1 + -3x + -2x2 Reorder the terms: 23 + -3x + -2x2 = 1 + -1 + 3x + -3x + 2x2 + -2x2 Combine like terms: 1 + -1 = 0 23 + -3x + -2x2 = 0 + 3x + -3x + 2x2 + -2x2 23 + -3x + -2x2 = 3x + -3x + 2x2 + -2x2 Combine like terms: 3x + -3x = 0 23 + -3x + -2x2 = 0 + 2x2 + -2x2 23 + -3x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 23 + -3x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11.5 + 1.5x + x2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 1.5x + 11.5 + x2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 1.5x + x2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 1.5x + x2 = 0 + 11.5 1.5x + x2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 1.5x + x2 = 11.5 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 11.5 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 11.5 + 0.5625 Combine like terms: 11.5 + 0.5625 = 12.0625 0.5625 + 1.5x + x2 = 12.0625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (x + 0.75) equal to 3.473110997 and -3.473110997.Subproblem 1
x + 0.75 = 3.473110997 Simplifying x + 0.75 = 3.473110997 Reorder the terms: 0.75 + x = 3.473110997 Solving 0.75 + x = 3.473110997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 3.473110997 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 3.473110997 + -0.75 x = 3.473110997 + -0.75 Combine like terms: 3.473110997 + -0.75 = 2.723110997 x = 2.723110997 Simplifying x = 2.723110997Subproblem 2
x + 0.75 = -3.473110997 Simplifying x + 0.75 = -3.473110997 Reorder the terms: 0.75 + x = -3.473110997 Solving 0.75 + x = -3.473110997 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -3.473110997 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -3.473110997 + -0.75 x = -3.473110997 + -0.75 Combine like terms: -3.473110997 + -0.75 = -4.223110997 x = -4.223110997 Simplifying x = -4.223110997Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.723110997, -4.223110997}
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