24x^2+180x+329=0

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Solution for 24x^2+180x+329=0 equation:


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 = -3.154880961

Subproblem 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 = -4.345119039

Solution

The solution to the problem is based on the solutions from the subproblems. x = {-3.154880961, -4.345119039}

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