21x^2-12x+1=0

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Solution for 21x^2-12x+1=0 equation:


Simplifying
21x2 + -12x + 1 = 0

Reorder the terms:
1 + -12x + 21x2 = 0

Solving
1 + -12x + 21x2 = 0

Solving for variable 'x'.

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

Divide each side by '21'.
0.04761904762 + -0.5714285714x + x2 = 0

Move the constant term to the right:

Add '-0.04761904762' to each side of the equation.
0.04761904762 + -0.5714285714x + -0.04761904762 + x2 = 0 + -0.04761904762

Reorder the terms:
0.04761904762 + -0.04761904762 + -0.5714285714x + x2 = 0 + -0.04761904762

Combine like terms: 0.04761904762 + -0.04761904762 = 0.00000000000
0.00000000000 + -0.5714285714x + x2 = 0 + -0.04761904762
-0.5714285714x + x2 = 0 + -0.04761904762

Combine like terms: 0 + -0.04761904762 = -0.04761904762
-0.5714285714x + x2 = -0.04761904762

The x term is -0.5714285714x.  Take half its coefficient (-0.2857142857).
Square it (0.08163265305) and add it to both sides.

Add '0.08163265305' to each side of the equation.
-0.5714285714x + 0.08163265305 + x2 = -0.04761904762 + 0.08163265305

Reorder the terms:
0.08163265305 + -0.5714285714x + x2 = -0.04761904762 + 0.08163265305

Combine like terms: -0.04761904762 + 0.08163265305 = 0.03401360543
0.08163265305 + -0.5714285714x + x2 = 0.03401360543

Factor a perfect square on the left side:
(x + -0.2857142857)(x + -0.2857142857) = 0.03401360543

Calculate the square root of the right side: 0.184427778

Break this problem into two subproblems by setting 
(x + -0.2857142857) equal to 0.184427778 and -0.184427778.

Subproblem 1

x + -0.2857142857 = 0.184427778 Simplifying x + -0.2857142857 = 0.184427778 Reorder the terms: -0.2857142857 + x = 0.184427778 Solving -0.2857142857 + x = 0.184427778 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = 0.184427778 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = 0.184427778 + 0.2857142857 x = 0.184427778 + 0.2857142857 Combine like terms: 0.184427778 + 0.2857142857 = 0.4701420637 x = 0.4701420637 Simplifying x = 0.4701420637

Subproblem 2

x + -0.2857142857 = -0.184427778 Simplifying x + -0.2857142857 = -0.184427778 Reorder the terms: -0.2857142857 + x = -0.184427778 Solving -0.2857142857 + x = -0.184427778 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + x = -0.184427778 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + x = -0.184427778 + 0.2857142857 x = -0.184427778 + 0.2857142857 Combine like terms: -0.184427778 + 0.2857142857 = 0.1012865077 x = 0.1012865077 Simplifying x = 0.1012865077

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.4701420637, 0.1012865077}

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