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Simplifying 24x2 + 180x + 329 = 0 Reorder the terms: 329 + 180x + 24x2 = 0 Solving 329 + 180x + 24x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. 13.70833333 + 7.5x + x2 = 0 Move the constant term to the right: Add '-13.70833333' to each side of the equation. 13.70833333 + 7.5x + -13.70833333 + x2 = 0 + -13.70833333 Reorder the terms: 13.70833333 + -13.70833333 + 7.5x + x2 = 0 + -13.70833333 Combine like terms: 13.70833333 + -13.70833333 = 0.00000000 0.00000000 + 7.5x + x2 = 0 + -13.70833333 7.5x + x2 = 0 + -13.70833333 Combine like terms: 0 + -13.70833333 = -13.70833333 7.5x + x2 = -13.70833333 The x term is 7.5x. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5x + 14.0625 + x2 = -13.70833333 + 14.0625 Reorder the terms: 14.0625 + 7.5x + x2 = -13.70833333 + 14.0625 Combine like terms: -13.70833333 + 14.0625 = 0.35416667 14.0625 + 7.5x + x2 = 0.35416667 Factor a perfect square on the left side: (x + 3.75)(x + 3.75) = 0.35416667 Calculate the square root of the right side: 0.595119039 Break this problem into two subproblems by setting (x + 3.75) equal to 0.595119039 and -0.595119039.Subproblem 1
x + 3.75 = 0.595119039 Simplifying x + 3.75 = 0.595119039 Reorder the terms: 3.75 + x = 0.595119039 Solving 3.75 + x = 0.595119039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = 0.595119039 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 0.595119039 + -3.75 x = 0.595119039 + -3.75 Combine like terms: 0.595119039 + -3.75 = -3.154880961 x = -3.154880961 Simplifying x = -3.154880961Subproblem 2
x + 3.75 = -0.595119039 Simplifying x + 3.75 = -0.595119039 Reorder the terms: 3.75 + x = -0.595119039 Solving 3.75 + x = -0.595119039 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = -0.595119039 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -0.595119039 + -3.75 x = -0.595119039 + -3.75 Combine like terms: -0.595119039 + -3.75 = -4.345119039 x = -4.345119039 Simplifying x = -4.345119039Solution
The solution to the problem is based on the solutions from the subproblems. x = {-3.154880961, -4.345119039}
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