22.23=2x^2+x

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Solution for 22.23=2x^2+x equation:


Simplifying
22.23 = 2x2 + x

Reorder the terms:
22.23 = x + 2x2

Solving
22.23 = x + 2x2

Solving for variable 'x'.

Reorder the terms:
22.23 + -1x + -2x2 = x + -1x + 2x2 + -2x2

Combine like terms: x + -1x = 0
22.23 + -1x + -2x2 = 0 + 2x2 + -2x2
22.23 + -1x + -2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
22.23 + -1x + -2x2 = 0

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

Divide each side by '-2'.
-11.115 + 0.5x + x2 = 0

Move the constant term to the right:

Add '11.115' to each side of the equation.
-11.115 + 0.5x + 11.115 + x2 = 0 + 11.115

Reorder the terms:
-11.115 + 11.115 + 0.5x + x2 = 0 + 11.115

Combine like terms: -11.115 + 11.115 = 0.000
0.000 + 0.5x + x2 = 0 + 11.115
0.5x + x2 = 0 + 11.115

Combine like terms: 0 + 11.115 = 11.115
0.5x + x2 = 11.115

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 11.115 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 11.115 + 0.0625

Combine like terms: 11.115 + 0.0625 = 11.1775
0.0625 + 0.5x + x2 = 11.1775

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 11.1775

Calculate the square root of the right side: 3.343276836

Break this problem into two subproblems by setting 
(x + 0.25) equal to 3.343276836 and -3.343276836.

Subproblem 1

x + 0.25 = 3.343276836 Simplifying x + 0.25 = 3.343276836 Reorder the terms: 0.25 + x = 3.343276836 Solving 0.25 + x = 3.343276836 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 3.343276836 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.343276836 + -0.25 x = 3.343276836 + -0.25 Combine like terms: 3.343276836 + -0.25 = 3.093276836 x = 3.093276836 Simplifying x = 3.093276836

Subproblem 2

x + 0.25 = -3.343276836 Simplifying x + 0.25 = -3.343276836 Reorder the terms: 0.25 + x = -3.343276836 Solving 0.25 + x = -3.343276836 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -3.343276836 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.343276836 + -0.25 x = -3.343276836 + -0.25 Combine like terms: -3.343276836 + -0.25 = -3.593276836 x = -3.593276836 Simplifying x = -3.593276836

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.093276836, -3.593276836}

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