3x^2+24x+1=-2

Solution for 3x^2+24x+1=-2 equation:

Simplifying
3x2 + 24x + 1 = -2

Reorder the terms:
1 + 24x + 3x2 = -2

Solving
1 + 24x + 3x2 = -2

Solving for variable 'x'.

Reorder the terms:
1 + 2 + 24x + 3x2 = -2 + 2

Combine like terms: 1 + 2 = 3
3 + 24x + 3x2 = -2 + 2

Combine like terms: -2 + 2 = 0
3 + 24x + 3x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(1 + 8x + x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(1 + 8x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 8x + x2 = 0

Solving
1 + 8x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 8x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 8x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 8x + x2 = 0 + -1
8x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
8x + x2 = -1

The x term is 8x.  Take half its coefficient (4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
8x + 16 + x2 = -1 + 16

Reorder the terms:
16 + 8x + x2 = -1 + 16

Combine like terms: -1 + 16 = 15
16 + 8x + x2 = 15

Factor a perfect square on the left side:
(x + 4)(x + 4) = 15

Calculate the square root of the right side: 3.872983346

Break this problem into two subproblems by setting 
(x + 4) equal to 3.872983346 and -3.872983346.

Subproblem 1
x + 4 = 3.872983346

Simplifying
x + 4 = 3.872983346

Reorder the terms:
4 + x = 3.872983346

Solving
4 + x = 3.872983346

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = 3.872983346 + -4

Combine like terms: 4 + -4 = 0
0 + x = 3.872983346 + -4
x = 3.872983346 + -4

Combine like terms: 3.872983346 + -4 = -0.127016654
x = -0.127016654

Simplifying
x = -0.127016654

Subproblem 2
x + 4 = -3.872983346

Simplifying
x + 4 = -3.872983346

Reorder the terms:
4 + x = -3.872983346

Solving
4 + x = -3.872983346

Solving for variable 'x'.

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

Add '-4' to each side of the equation.
4 + -4 + x = -3.872983346 + -4

Combine like terms: 4 + -4 = 0
0 + x = -3.872983346 + -4
x = -3.872983346 + -4

Combine like terms: -3.872983346 + -4 = -7.872983346
x = -7.872983346

Simplifying
x = -7.872983346

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.127016654, -7.872983346}
Solution
x = {-0.127016654, -7.872983346}