7x^2+53x+24=0

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Solution for 7x^2+53x+24=0 equation: 


Simplifying
7x2 + 53x + 24 = 0

Reorder the terms:
24 + 53x + 7x2 = 0

Solving
24 + 53x + 7x2 = 0

Solving for variable 'x'.

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

Divide each side by '7'.
3.428571429 + 7.571428571x + x2 = 0

Move the constant term to the right:

Add '-3.428571429' to each side of the equation.
3.428571429 + 7.571428571x + -3.428571429 + x2 = 0 + -3.428571429

Reorder the terms:
3.428571429 + -3.428571429 + 7.571428571x + x2 = 0 + -3.428571429

Combine like terms: 3.428571429 + -3.428571429 = 0.000000000
0.000000000 + 7.571428571x + x2 = 0 + -3.428571429
7.571428571x + x2 = 0 + -3.428571429

Combine like terms: 0 + -3.428571429 = -3.428571429
7.571428571x + x2 = -3.428571429

The x term is 7.571428571x.  Take half its coefficient (3.785714286).
Square it (14.33163266) and add it to both sides.

Add '14.33163266' to each side of the equation.
7.571428571x + 14.33163266 + x2 = -3.428571429 + 14.33163266

Reorder the terms:
14.33163266 + 7.571428571x + x2 = -3.428571429 + 14.33163266

Combine like terms: -3.428571429 + 14.33163266 = 10.903061231
14.33163266 + 7.571428571x + x2 = 10.903061231

Factor a perfect square on the left side:
(x + 3.785714286)(x + 3.785714286) = 10.903061231

Calculate the square root of the right side: 3.301978381

Break this problem into two subproblems by setting 
(x + 3.785714286) equal to 3.301978381 and -3.301978381.


Subproblem 1
x + 3.785714286 = 3.301978381

Simplifying
x + 3.785714286 = 3.301978381

Reorder the terms:
3.785714286 + x = 3.301978381

Solving
3.785714286 + x = 3.301978381

Solving for variable 'x'.

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

Add '-3.785714286' to each side of the equation.
3.785714286 + -3.785714286 + x = 3.301978381 + -3.785714286

Combine like terms: 3.785714286 + -3.785714286 = 0.000000000
0.000000000 + x = 3.301978381 + -3.785714286
x = 3.301978381 + -3.785714286

Combine like terms: 3.301978381 + -3.785714286 = -0.483735905
x = -0.483735905

Simplifying
x = -0.483735905


Subproblem 2
x + 3.785714286 = -3.301978381

Simplifying
x + 3.785714286 = -3.301978381

Reorder the terms:
3.785714286 + x = -3.301978381

Solving
3.785714286 + x = -3.301978381

Solving for variable 'x'.

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

Add '-3.785714286' to each side of the equation.
3.785714286 + -3.785714286 + x = -3.301978381 + -3.785714286

Combine like terms: 3.785714286 + -3.785714286 = 0.000000000
0.000000000 + x = -3.301978381 + -3.785714286
x = -3.301978381 + -3.785714286

Combine like terms: -3.301978381 + -3.785714286 = -7.087692667
x = -7.087692667

Simplifying
x = -7.087692667


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.483735905, -7.087692667}

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