4x^2+27x^2-24x-120=0

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Solution for 4x^2+27x^2-24x-120=0 equation:


Simplifying
4x2 + 27x2 + -24x + -120 = 0

Reorder the terms:
-120 + -24x + 4x2 + 27x2 = 0

Combine like terms: 4x2 + 27x2 = 31x2
-120 + -24x + 31x2 = 0

Solving
-120 + -24x + 31x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
31 the coefficient of the squared term: 

Divide each side by '31'.
-3.870967742 + -0.7741935484x + x2 = 0

Move the constant term to the right:

Add '3.870967742' to each side of the equation.
-3.870967742 + -0.7741935484x + 3.870967742 + x2 = 0 + 3.870967742

Reorder the terms:
-3.870967742 + 3.870967742 + -0.7741935484x + x2 = 0 + 3.870967742

Combine like terms: -3.870967742 + 3.870967742 = 0.000000000
0.000000000 + -0.7741935484x + x2 = 0 + 3.870967742
-0.7741935484x + x2 = 0 + 3.870967742

Combine like terms: 0 + 3.870967742 = 3.870967742
-0.7741935484x + x2 = 3.870967742

The x term is -0.7741935484x.  Take half its coefficient (-0.3870967742).
Square it (0.1498439126) and add it to both sides.

Add '0.1498439126' to each side of the equation.
-0.7741935484x + 0.1498439126 + x2 = 3.870967742 + 0.1498439126

Reorder the terms:
0.1498439126 + -0.7741935484x + x2 = 3.870967742 + 0.1498439126

Combine like terms: 3.870967742 + 0.1498439126 = 4.0208116546
0.1498439126 + -0.7741935484x + x2 = 4.0208116546

Factor a perfect square on the left side:
(x + -0.3870967742)(x + -0.3870967742) = 4.0208116546

Calculate the square root of the right side: 2.005196164

Break this problem into two subproblems by setting 
(x + -0.3870967742) equal to 2.005196164 and -2.005196164.

Subproblem 1

x + -0.3870967742 = 2.005196164 Simplifying x + -0.3870967742 = 2.005196164 Reorder the terms: -0.3870967742 + x = 2.005196164 Solving -0.3870967742 + x = 2.005196164 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3870967742' to each side of the equation. -0.3870967742 + 0.3870967742 + x = 2.005196164 + 0.3870967742 Combine like terms: -0.3870967742 + 0.3870967742 = 0.0000000000 0.0000000000 + x = 2.005196164 + 0.3870967742 x = 2.005196164 + 0.3870967742 Combine like terms: 2.005196164 + 0.3870967742 = 2.3922929382 x = 2.3922929382 Simplifying x = 2.3922929382

Subproblem 2

x + -0.3870967742 = -2.005196164 Simplifying x + -0.3870967742 = -2.005196164 Reorder the terms: -0.3870967742 + x = -2.005196164 Solving -0.3870967742 + x = -2.005196164 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3870967742' to each side of the equation. -0.3870967742 + 0.3870967742 + x = -2.005196164 + 0.3870967742 Combine like terms: -0.3870967742 + 0.3870967742 = 0.0000000000 0.0000000000 + x = -2.005196164 + 0.3870967742 x = -2.005196164 + 0.3870967742 Combine like terms: -2.005196164 + 0.3870967742 = -1.6180993898 x = -1.6180993898 Simplifying x = -1.6180993898

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.3922929382, -1.6180993898}

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