3x^2+53x+24=0

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Solution for 3x^2+53x+24=0 equation: 


Simplifying
3x2 + 53x + 24 = 0

Reorder the terms:
24 + 53x + 3x2 = 0

Solving
24 + 53x + 3x2 = 0

Solving for variable 'x'.

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

Divide each side by '3'.
8 + 17.66666667x + x2 = 0

Move the constant term to the right:

Add '-8' to each side of the equation.
8 + 17.66666667x + -8 + x2 = 0 + -8

Reorder the terms:
8 + -8 + 17.66666667x + x2 = 0 + -8

Combine like terms: 8 + -8 = 0
0 + 17.66666667x + x2 = 0 + -8
17.66666667x + x2 = 0 + -8

Combine like terms: 0 + -8 = -8
17.66666667x + x2 = -8

The x term is 17.66666667x.  Take half its coefficient (8.833333335).
Square it (78.02777781) and add it to both sides.

Add '78.02777781' to each side of the equation.
17.66666667x + 78.02777781 + x2 = -8 + 78.02777781

Reorder the terms:
78.02777781 + 17.66666667x + x2 = -8 + 78.02777781

Combine like terms: -8 + 78.02777781 = 70.02777781
78.02777781 + 17.66666667x + x2 = 70.02777781

Factor a perfect square on the left side:
(x + 8.833333335)(x + 8.833333335) = 70.02777781

Calculate the square root of the right side: 8.368260142

Break this problem into two subproblems by setting 
(x + 8.833333335) equal to 8.368260142 and -8.368260142.


Subproblem 1
x + 8.833333335 = 8.368260142

Simplifying
x + 8.833333335 = 8.368260142

Reorder the terms:
8.833333335 + x = 8.368260142

Solving
8.833333335 + x = 8.368260142

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-8.833333335' to each side of the equation.
8.833333335 + -8.833333335 + x = 8.368260142 + -8.833333335

Combine like terms: 8.833333335 + -8.833333335 = 0.000000000
0.000000000 + x = 8.368260142 + -8.833333335
x = 8.368260142 + -8.833333335

Combine like terms: 8.368260142 + -8.833333335 = -0.465073193
x = -0.465073193

Simplifying
x = -0.465073193


Subproblem 2
x + 8.833333335 = -8.368260142

Simplifying
x + 8.833333335 = -8.368260142

Reorder the terms:
8.833333335 + x = -8.368260142

Solving
8.833333335 + x = -8.368260142

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-8.833333335' to each side of the equation.
8.833333335 + -8.833333335 + x = -8.368260142 + -8.833333335

Combine like terms: 8.833333335 + -8.833333335 = 0.000000000
0.000000000 + x = -8.368260142 + -8.833333335
x = -8.368260142 + -8.833333335

Combine like terms: -8.368260142 + -8.833333335 = -17.201593477
x = -17.201593477

Simplifying
x = -17.201593477


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.465073193, -17.201593477}

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