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Simplifying (2x + 5)(2x + 1) = 180 Reorder the terms: (5 + 2x)(2x + 1) = 180 Reorder the terms: (5 + 2x)(1 + 2x) = 180 Multiply (5 + 2x) * (1 + 2x) (5(1 + 2x) + 2x * (1 + 2x)) = 180 ((1 * 5 + 2x * 5) + 2x * (1 + 2x)) = 180 ((5 + 10x) + 2x * (1 + 2x)) = 180 (5 + 10x + (1 * 2x + 2x * 2x)) = 180 (5 + 10x + (2x + 4x2)) = 180 Combine like terms: 10x + 2x = 12x (5 + 12x + 4x2) = 180 Solving 5 + 12x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: 5 + -180 + 12x + 4x2 = 180 + -180 Combine like terms: 5 + -180 = -175 -175 + 12x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -175 + 12x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -43.75 + 3x + x2 = 0 Move the constant term to the right: Add '43.75' to each side of the equation. -43.75 + 3x + 43.75 + x2 = 0 + 43.75 Reorder the terms: -43.75 + 43.75 + 3x + x2 = 0 + 43.75 Combine like terms: -43.75 + 43.75 = 0.00 0.00 + 3x + x2 = 0 + 43.75 3x + x2 = 0 + 43.75 Combine like terms: 0 + 43.75 = 43.75 3x + x2 = 43.75 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 43.75 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 43.75 + 2.25 Combine like terms: 43.75 + 2.25 = 46 2.25 + 3x + x2 = 46 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 46 Calculate the square root of the right side: 6.782329983 Break this problem into two subproblems by setting (x + 1.5) equal to 6.782329983 and -6.782329983.Subproblem 1
x + 1.5 = 6.782329983 Simplifying x + 1.5 = 6.782329983 Reorder the terms: 1.5 + x = 6.782329983 Solving 1.5 + x = 6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 6.782329983 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 6.782329983 + -1.5 x = 6.782329983 + -1.5 Combine like terms: 6.782329983 + -1.5 = 5.282329983 x = 5.282329983 Simplifying x = 5.282329983Subproblem 2
x + 1.5 = -6.782329983 Simplifying x + 1.5 = -6.782329983 Reorder the terms: 1.5 + x = -6.782329983 Solving 1.5 + x = -6.782329983 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -6.782329983 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -6.782329983 + -1.5 x = -6.782329983 + -1.5 Combine like terms: -6.782329983 + -1.5 = -8.282329983 x = -8.282329983 Simplifying x = -8.282329983Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.282329983, -8.282329983}
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