4x^2+6x+2=90000

Solution for 4x^2+6x+2=90000 equation:

Simplifying
4x2 + 6x + 2 = 90000

Reorder the terms:
2 + 6x + 4x2 = 90000

Solving
2 + 6x + 4x2 = 90000

Solving for variable 'x'.

Reorder the terms:
2 + -90000 + 6x + 4x2 = 90000 + -90000

Combine like terms: 2 + -90000 = -89998
-89998 + 6x + 4x2 = 90000 + -90000

Combine like terms: 90000 + -90000 = 0
-89998 + 6x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-44999 + 3x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-44999 + 3x + 2x2)' equal to zero and attempt to solve:

Simplifying
-44999 + 3x + 2x2 = 0

Solving
-44999 + 3x + 2x2 = 0

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

Divide each side by '2'.
-22499.5 + 1.5x + x2 = 0

Move the constant term to the right:

Add '22499.5' to each side of the equation.
-22499.5 + 1.5x + 22499.5 + x2 = 0 + 22499.5

Reorder the terms:
-22499.5 + 22499.5 + 1.5x + x2 = 0 + 22499.5

Combine like terms: -22499.5 + 22499.5 = 0.0
0.0 + 1.5x + x2 = 0 + 22499.5
1.5x + x2 = 0 + 22499.5

Combine like terms: 0 + 22499.5 = 22499.5
1.5x + x2 = 22499.5

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = 22499.5 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = 22499.5 + 0.5625

Combine like terms: 22499.5 + 0.5625 = 22500.0625
0.5625 + 1.5x + x2 = 22500.0625

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 22500.0625

Calculate the square root of the right side: 150.000208333

Break this problem into two subproblems by setting 
(x + 0.75) equal to 150.000208333 and -150.000208333.

Subproblem 1
x + 0.75 = 150.000208333

Simplifying
x + 0.75 = 150.000208333

Reorder the terms:
0.75 + x = 150.000208333

Solving
0.75 + x = 150.000208333

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 150.000208333 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 150.000208333 + -0.75
x = 150.000208333 + -0.75

Combine like terms: 150.000208333 + -0.75 = 149.250208333
x = 149.250208333

Simplifying
x = 149.250208333

Subproblem 2
x + 0.75 = -150.000208333

Simplifying
x + 0.75 = -150.000208333

Reorder the terms:
0.75 + x = -150.000208333

Solving
0.75 + x = -150.000208333

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -150.000208333 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -150.000208333 + -0.75
x = -150.000208333 + -0.75

Combine like terms: -150.000208333 + -0.75 = -150.750208333
x = -150.750208333

Simplifying
x = -150.750208333

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {149.250208333, -150.750208333}
Solution
x = {149.250208333, -150.750208333}