21x^2-61x+6=0

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Solution for 21x^2-61x+6=0 equation:


Simplifying
21x2 + -61x + 6 = 0

Reorder the terms:
6 + -61x + 21x2 = 0

Solving
6 + -61x + 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'.
0.2857142857 + -2.904761905x + x2 = 0

Move the constant term to the right:

Add '-0.2857142857' to each side of the equation.
0.2857142857 + -2.904761905x + -0.2857142857 + x2 = 0 + -0.2857142857

Reorder the terms:
0.2857142857 + -0.2857142857 + -2.904761905x + x2 = 0 + -0.2857142857

Combine like terms: 0.2857142857 + -0.2857142857 = 0.0000000000
0.0000000000 + -2.904761905x + x2 = 0 + -0.2857142857
-2.904761905x + x2 = 0 + -0.2857142857

Combine like terms: 0 + -0.2857142857 = -0.2857142857
-2.904761905x + x2 = -0.2857142857

The x term is -2.904761905x.  Take half its coefficient (-1.452380953).
Square it (2.109410433) and add it to both sides.

Add '2.109410433' to each side of the equation.
-2.904761905x + 2.109410433 + x2 = -0.2857142857 + 2.109410433

Reorder the terms:
2.109410433 + -2.904761905x + x2 = -0.2857142857 + 2.109410433

Combine like terms: -0.2857142857 + 2.109410433 = 1.8236961473
2.109410433 + -2.904761905x + x2 = 1.8236961473

Factor a perfect square on the left side:
(x + -1.452380953)(x + -1.452380953) = 1.8236961473

Calculate the square root of the right side: 1.350442945

Break this problem into two subproblems by setting 
(x + -1.452380953) equal to 1.350442945 and -1.350442945.

Subproblem 1

x + -1.452380953 = 1.350442945 Simplifying x + -1.452380953 = 1.350442945 Reorder the terms: -1.452380953 + x = 1.350442945 Solving -1.452380953 + x = 1.350442945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.452380953' to each side of the equation. -1.452380953 + 1.452380953 + x = 1.350442945 + 1.452380953 Combine like terms: -1.452380953 + 1.452380953 = 0.000000000 0.000000000 + x = 1.350442945 + 1.452380953 x = 1.350442945 + 1.452380953 Combine like terms: 1.350442945 + 1.452380953 = 2.802823898 x = 2.802823898 Simplifying x = 2.802823898

Subproblem 2

x + -1.452380953 = -1.350442945 Simplifying x + -1.452380953 = -1.350442945 Reorder the terms: -1.452380953 + x = -1.350442945 Solving -1.452380953 + x = -1.350442945 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.452380953' to each side of the equation. -1.452380953 + 1.452380953 + x = -1.350442945 + 1.452380953 Combine like terms: -1.452380953 + 1.452380953 = 0.000000000 0.000000000 + x = -1.350442945 + 1.452380953 x = -1.350442945 + 1.452380953 Combine like terms: -1.350442945 + 1.452380953 = 0.101938008 x = 0.101938008 Simplifying x = 0.101938008

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.802823898, 0.101938008}

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