x+3=224x^2

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Solution for x+3=224x^2 equation:


Simplifying
x + 3 = 224x2

Reorder the terms:
3 + x = 224x2

Solving
3 + x = 224x2

Solving for variable 'x'.

Combine like terms: 224x2 + -224x2 = 0
3 + x + -224x2 = 0

Begin completing the square.  Divide all terms by
-224 the coefficient of the squared term: 

Divide each side by '-224'.
-0.01339285714 + -0.004464285714x + x2 = 0

Move the constant term to the right:

Add '0.01339285714' to each side of the equation.
-0.01339285714 + -0.004464285714x + 0.01339285714 + x2 = 0 + 0.01339285714

Reorder the terms:
-0.01339285714 + 0.01339285714 + -0.004464285714x + x2 = 0 + 0.01339285714

Combine like terms: -0.01339285714 + 0.01339285714 = 0.00000000000
0.00000000000 + -0.004464285714x + x2 = 0 + 0.01339285714
-0.004464285714x + x2 = 0 + 0.01339285714

Combine like terms: 0 + 0.01339285714 = 0.01339285714
-0.004464285714x + x2 = 0.01339285714

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.
-0.004464285714x + 0.25 + x2 = 0.01339285714 + 0.25

Reorder the terms:
0.25 + -0.004464285714x + x2 = 0.01339285714 + 0.25

Combine like terms: 0.01339285714 + 0.25 = 0.26339285714
0.25 + -0.004464285714x + x2 = 0.26339285714

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 0.26339285714

Calculate the square root of the right side: 0.513218138

Break this problem into two subproblems by setting 
(x + 0.5) equal to 0.513218138 and -0.513218138.

Subproblem 1

x + 0.5 = 0.513218138 Simplifying x + 0.5 = 0.513218138 Reorder the terms: 0.5 + x = 0.513218138 Solving 0.5 + x = 0.513218138 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 = 0.513218138 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.513218138 + -0.5 x = 0.513218138 + -0.5 Combine like terms: 0.513218138 + -0.5 = 0.013218138 x = 0.013218138 Simplifying x = 0.013218138

Subproblem 2

x + 0.5 = -0.513218138 Simplifying x + 0.5 = -0.513218138 Reorder the terms: 0.5 + x = -0.513218138 Solving 0.5 + x = -0.513218138 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 = -0.513218138 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.513218138 + -0.5 x = -0.513218138 + -0.5 Combine like terms: -0.513218138 + -0.5 = -1.013218138 x = -1.013218138 Simplifying x = -1.013218138

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.013218138, -1.013218138}

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