21x^2-11x+1=0

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Solution for 21x^2-11x+1=0 equation:


Simplifying
21x2 + -11x + 1 = 0

Reorder the terms:
1 + -11x + 21x2 = 0

Solving
1 + -11x + 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 + -0.5238095238x + x2 = 0

Move the constant term to the right:

Add '-0.04761904762' to each side of the equation.
0.04761904762 + -0.5238095238x + -0.04761904762 + x2 = 0 + -0.04761904762

Reorder the terms:
0.04761904762 + -0.04761904762 + -0.5238095238x + x2 = 0 + -0.04761904762

Combine like terms: 0.04761904762 + -0.04761904762 = 0.00000000000
0.00000000000 + -0.5238095238x + x2 = 0 + -0.04761904762
-0.5238095238x + x2 = 0 + -0.04761904762

Combine like terms: 0 + -0.04761904762 = -0.04761904762
-0.5238095238x + x2 = -0.04761904762

The x term is -0.5238095238x.  Take half its coefficient (-0.2619047619).
Square it (0.06859410431) and add it to both sides.

Add '0.06859410431' to each side of the equation.
-0.5238095238x + 0.06859410431 + x2 = -0.04761904762 + 0.06859410431

Reorder the terms:
0.06859410431 + -0.5238095238x + x2 = -0.04761904762 + 0.06859410431

Combine like terms: -0.04761904762 + 0.06859410431 = 0.02097505669
0.06859410431 + -0.5238095238x + x2 = 0.02097505669

Factor a perfect square on the left side:
(x + -0.2619047619)(x + -0.2619047619) = 0.02097505669

Calculate the square root of the right side: 0.144827679

Break this problem into two subproblems by setting 
(x + -0.2619047619) equal to 0.144827679 and -0.144827679.

Subproblem 1

x + -0.2619047619 = 0.144827679 Simplifying x + -0.2619047619 = 0.144827679 Reorder the terms: -0.2619047619 + x = 0.144827679 Solving -0.2619047619 + x = 0.144827679 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2619047619' to each side of the equation. -0.2619047619 + 0.2619047619 + x = 0.144827679 + 0.2619047619 Combine like terms: -0.2619047619 + 0.2619047619 = 0.0000000000 0.0000000000 + x = 0.144827679 + 0.2619047619 x = 0.144827679 + 0.2619047619 Combine like terms: 0.144827679 + 0.2619047619 = 0.4067324409 x = 0.4067324409 Simplifying x = 0.4067324409

Subproblem 2

x + -0.2619047619 = -0.144827679 Simplifying x + -0.2619047619 = -0.144827679 Reorder the terms: -0.2619047619 + x = -0.144827679 Solving -0.2619047619 + x = -0.144827679 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.2619047619' to each side of the equation. -0.2619047619 + 0.2619047619 + x = -0.144827679 + 0.2619047619 Combine like terms: -0.2619047619 + 0.2619047619 = 0.0000000000 0.0000000000 + x = -0.144827679 + 0.2619047619 x = -0.144827679 + 0.2619047619 Combine like terms: -0.144827679 + 0.2619047619 = 0.1170770829 x = 0.1170770829 Simplifying x = 0.1170770829

Solution

The solution to the problem is based on the solutions from the subproblems. x = {0.4067324409, 0.1170770829}

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