24x^2+168x+282=0

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Solution for 24x^2+168x+282=0 equation: 


Simplifying
24x2 + 168x + 282 = 0

Reorder the terms:
282 + 168x + 24x2 = 0

Solving
282 + 168x + 24x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '6'.
6(47 + 28x + 4x2) = 0

Ignore the factor 6.

Subproblem 1
Set the factor '(47 + 28x + 4x2)' equal to zero and attempt to solve:

Simplifying
47 + 28x + 4x2 = 0

Solving
47 + 28x + 4x2 = 0

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

Divide each side by '4'.
11.75 + 7x + x2 = 0

Move the constant term to the right:

Add '-11.75' to each side of the equation.
11.75 + 7x + -11.75 + x2 = 0 + -11.75

Reorder the terms:
11.75 + -11.75 + 7x + x2 = 0 + -11.75

Combine like terms: 11.75 + -11.75 = 0.00
0.00 + 7x + x2 = 0 + -11.75
7x + x2 = 0 + -11.75

Combine like terms: 0 + -11.75 = -11.75
7x + x2 = -11.75

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -11.75 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -11.75 + 12.25

Combine like terms: -11.75 + 12.25 = 0.5
12.25 + 7x + x2 = 0.5

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + 3.5) equal to 0.707106781 and -0.707106781.


Subproblem 1
x + 3.5 = 0.707106781

Simplifying
x + 3.5 = 0.707106781

Reorder the terms:
3.5 + x = 0.707106781

Solving
3.5 + x = 0.707106781

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 0.707106781 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 0.707106781 + -3.5
x = 0.707106781 + -3.5

Combine like terms: 0.707106781 + -3.5 = -2.792893219
x = -2.792893219

Simplifying
x = -2.792893219


Subproblem 2
x + 3.5 = -0.707106781

Simplifying
x + 3.5 = -0.707106781

Reorder the terms:
3.5 + x = -0.707106781

Solving
3.5 + x = -0.707106781

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -0.707106781 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -0.707106781 + -3.5
x = -0.707106781 + -3.5

Combine like terms: -0.707106781 + -3.5 = -4.207106781
x = -4.207106781

Simplifying
x = -4.207106781


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.792893219, -4.207106781}
Solution
x = {-2.792893219, -4.207106781}

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