2x^2-62.35x+481=406

Solution for 2x^2-62.35x+481=406 equation:

Simplifying
2x2 + -62.35x + 481 = 406

Reorder the terms:
481 + -62.35x + 2x2 = 406

Solving
481 + -62.35x + 2x2 = 406

Solving for variable 'x'.

Reorder the terms:
481 + -406 + -62.35x + 2x2 = 406 + -406

Combine like terms: 481 + -406 = 75
75 + -62.35x + 2x2 = 406 + -406

Combine like terms: 406 + -406 = 0
75 + -62.35x + 2x2 = 0

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

Divide each side by '2'.
37.5 + -31.175x + x2 = 0

Move the constant term to the right:

Add '-37.5' to each side of the equation.
37.5 + -31.175x + -37.5 + x2 = 0 + -37.5

Reorder the terms:
37.5 + -37.5 + -31.175x + x2 = 0 + -37.5

Combine like terms: 37.5 + -37.5 = 0.0
0.0 + -31.175x + x2 = 0 + -37.5
-31.175x + x2 = 0 + -37.5

Combine like terms: 0 + -37.5 = -37.5
-31.175x + x2 = -37.5

The x term is -31.175x.  Take half its coefficient (-15.5875).
Square it (242.9701563) and add it to both sides.

Add '242.9701563' to each side of the equation.
-31.175x + 242.9701563 + x2 = -37.5 + 242.9701563

Reorder the terms:
242.9701563 + -31.175x + x2 = -37.5 + 242.9701563

Combine like terms: -37.5 + 242.9701563 = 205.4701563
242.9701563 + -31.175x + x2 = 205.4701563

Factor a perfect square on the left side:
(x + -15.5875)(x + -15.5875) = 205.4701563

Calculate the square root of the right side: 14.33423023

Break this problem into two subproblems by setting 
(x + -15.5875) equal to 14.33423023 and -14.33423023.

Subproblem 1
x + -15.5875 = 14.33423023

Simplifying
x + -15.5875 = 14.33423023

Reorder the terms:
-15.5875 + x = 14.33423023

Solving
-15.5875 + x = 14.33423023

Solving for variable 'x'.

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

Add '15.5875' to each side of the equation.
-15.5875 + 15.5875 + x = 14.33423023 + 15.5875

Combine like terms: -15.5875 + 15.5875 = 0.0000
0.0000 + x = 14.33423023 + 15.5875
x = 14.33423023 + 15.5875

Combine like terms: 14.33423023 + 15.5875 = 29.92173023
x = 29.92173023

Simplifying
x = 29.92173023

Subproblem 2
x + -15.5875 = -14.33423023

Simplifying
x + -15.5875 = -14.33423023

Reorder the terms:
-15.5875 + x = -14.33423023

Solving
-15.5875 + x = -14.33423023

Solving for variable 'x'.

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

Add '15.5875' to each side of the equation.
-15.5875 + 15.5875 + x = -14.33423023 + 15.5875

Combine like terms: -15.5875 + 15.5875 = 0.0000
0.0000 + x = -14.33423023 + 15.5875
x = -14.33423023 + 15.5875

Combine like terms: -14.33423023 + 15.5875 = 1.25326977
x = 1.25326977

Simplifying
x = 1.25326977

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {29.92173023, 1.25326977}