2x^2+1+2x=49

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Solution for 2x^2+1+2x=49 equation: 


Simplifying
2x2 + 1 + 2x = 49

Reorder the terms:
1 + 2x + 2x2 = 49

Solving
1 + 2x + 2x2 = 49

Solving for variable 'x'.

Reorder the terms:
1 + -49 + 2x + 2x2 = 49 + -49

Combine like terms: 1 + -49 = -48
-48 + 2x + 2x2 = 49 + -49

Combine like terms: 49 + -49 = 0
-48 + 2x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-24 + x + x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-24 + x + x2)' equal to zero and attempt to solve:

Simplifying
-24 + x + x2 = 0

Solving
-24 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + x + 24 + x2 = 0 + 24

Reorder the terms:
-24 + 24 + x + x2 = 0 + 24

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

Combine like terms: 0 + 24 = 24
x + x2 = 24

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 24 + 0.25

Reorder the terms:
0.25 + x + x2 = 24 + 0.25

Combine like terms: 24 + 0.25 = 24.25
0.25 + x + x2 = 24.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 24.25

Calculate the square root of the right side: 4.924428901

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.924428901 and -4.924428901.


Subproblem 1
x + 0.5 = 4.924428901

Simplifying
x + 0.5 = 4.924428901

Reorder the terms:
0.5 + x = 4.924428901

Solving
0.5 + x = 4.924428901

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.924428901 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.924428901 + -0.5
x = 4.924428901 + -0.5

Combine like terms: 4.924428901 + -0.5 = 4.424428901
x = 4.424428901

Simplifying
x = 4.424428901


Subproblem 2
x + 0.5 = -4.924428901

Simplifying
x + 0.5 = -4.924428901

Reorder the terms:
0.5 + x = -4.924428901

Solving
0.5 + x = -4.924428901

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.924428901 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.924428901 + -0.5
x = -4.924428901 + -0.5

Combine like terms: -4.924428901 + -0.5 = -5.424428901
x = -5.424428901

Simplifying
x = -5.424428901


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.424428901, -5.424428901}
Solution
x = {4.424428901, -5.424428901}

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