11x=2x^2+3

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Solution for 11x=2x^2+3 equation: 


Simplifying
11x = 2x2 + 3

Reorder the terms:
11x = 3 + 2x2

Solving
11x = 3 + 2x2

Solving for variable 'x'.

Reorder the terms:
-3 + 11x + -2x2 = 3 + 2x2 + -3 + -2x2

Reorder the terms:
-3 + 11x + -2x2 = 3 + -3 + 2x2 + -2x2

Combine like terms: 3 + -3 = 0
-3 + 11x + -2x2 = 0 + 2x2 + -2x2
-3 + 11x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-3 + 11x + -2x2 = 0

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

Divide each side by '-2'.
1.5 + -5.5x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -5.5x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -5.5x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -5.5x + x2 = 0 + -1.5
-5.5x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-5.5x + x2 = -1.5

The x term is -5.5x.  Take half its coefficient (-2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
-5.5x + 7.5625 + x2 = -1.5 + 7.5625

Reorder the terms:
7.5625 + -5.5x + x2 = -1.5 + 7.5625

Combine like terms: -1.5 + 7.5625 = 6.0625
7.5625 + -5.5x + x2 = 6.0625

Factor a perfect square on the left side:
(x + -2.75)(x + -2.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(x + -2.75) equal to 2.46221445 and -2.46221445.


Subproblem 1
x + -2.75 = 2.46221445

Simplifying
x + -2.75 = 2.46221445

Reorder the terms:
-2.75 + x = 2.46221445

Solving
-2.75 + x = 2.46221445

Solving for variable 'x'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + x = 2.46221445 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + x = 2.46221445 + 2.75
x = 2.46221445 + 2.75

Combine like terms: 2.46221445 + 2.75 = 5.21221445
x = 5.21221445

Simplifying
x = 5.21221445


Subproblem 2
x + -2.75 = -2.46221445

Simplifying
x + -2.75 = -2.46221445

Reorder the terms:
-2.75 + x = -2.46221445

Solving
-2.75 + x = -2.46221445

Solving for variable 'x'.

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

Add '2.75' to each side of the equation.
-2.75 + 2.75 + x = -2.46221445 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + x = -2.46221445 + 2.75
x = -2.46221445 + 2.75

Combine like terms: -2.46221445 + 2.75 = 0.28778555
x = 0.28778555

Simplifying
x = 0.28778555


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.21221445, 0.28778555}

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