22x^2+x=24

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Solution for 22x^2+x=24 equation:


Simplifying
22x2 + x = 24

Reorder the terms:
x + 22x2 = 24

Solving
x + 22x2 = 24

Solving for variable 'x'.

Reorder the terms:
-24 + x + 22x2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + x + 22x2 = 0

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

Divide each side by '22'.
-1.090909091 + 0.04545454545x + x2 = 0

Move the constant term to the right:

Add '1.090909091' to each side of the equation.
-1.090909091 + 0.04545454545x + 1.090909091 + x2 = 0 + 1.090909091

Reorder the terms:
-1.090909091 + 1.090909091 + 0.04545454545x + x2 = 0 + 1.090909091

Combine like terms: -1.090909091 + 1.090909091 = 0.000000000
0.000000000 + 0.04545454545x + x2 = 0 + 1.090909091
0.04545454545x + x2 = 0 + 1.090909091

Combine like terms: 0 + 1.090909091 = 1.090909091
0.04545454545x + x2 = 1.090909091

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.
0.04545454545x + 0.25 + x2 = 1.090909091 + 0.25

Reorder the terms:
0.25 + 0.04545454545x + x2 = 1.090909091 + 0.25

Combine like terms: 1.090909091 + 0.25 = 1.340909091
0.25 + 0.04545454545x + x2 = 1.340909091

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.340909091

Calculate the square root of the right side: 1.157976291

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.157976291 and -1.157976291.

Subproblem 1

x + 0.5 = 1.157976291 Simplifying x + 0.5 = 1.157976291 Reorder the terms: 0.5 + x = 1.157976291 Solving 0.5 + x = 1.157976291 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 = 1.157976291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.157976291 + -0.5 x = 1.157976291 + -0.5 Combine like terms: 1.157976291 + -0.5 = 0.657976291 x = 0.657976291 Simplifying x = 0.657976291

Subproblem 2

x + 0.5 = -1.157976291 Simplifying x + 0.5 = -1.157976291 Reorder the terms: 0.5 + x = -1.157976291 Solving 0.5 + x = -1.157976291 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 = -1.157976291 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.157976291 + -0.5 x = -1.157976291 + -0.5 Combine like terms: -1.157976291 + -0.5 = -1.657976291 x = -1.657976291 Simplifying x = -1.657976291

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.657976291, -1.657976291}

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