x^2-21x+109=0

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Solution for x^2-21x+109=0 equation:


Simplifying
x2 + -21x + 109 = 0

Reorder the terms:
109 + -21x + x2 = 0

Solving
109 + -21x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-109' to each side of the equation.
109 + -21x + -109 + x2 = 0 + -109

Reorder the terms:
109 + -109 + -21x + x2 = 0 + -109

Combine like terms: 109 + -109 = 0
0 + -21x + x2 = 0 + -109
-21x + x2 = 0 + -109

Combine like terms: 0 + -109 = -109
-21x + x2 = -109

The x term is -21x.  Take half its coefficient (-10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
-21x + 110.25 + x2 = -109 + 110.25

Reorder the terms:
110.25 + -21x + x2 = -109 + 110.25

Combine like terms: -109 + 110.25 = 1.25
110.25 + -21x + x2 = 1.25

Factor a perfect square on the left side:
(x + -10.5)(x + -10.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + -10.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

x + -10.5 = 1.118033989 Simplifying x + -10.5 = 1.118033989 Reorder the terms: -10.5 + x = 1.118033989 Solving -10.5 + x = 1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = 1.118033989 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = 1.118033989 + 10.5 x = 1.118033989 + 10.5 Combine like terms: 1.118033989 + 10.5 = 11.618033989 x = 11.618033989 Simplifying x = 11.618033989

Subproblem 2

x + -10.5 = -1.118033989 Simplifying x + -10.5 = -1.118033989 Reorder the terms: -10.5 + x = -1.118033989 Solving -10.5 + x = -1.118033989 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '10.5' to each side of the equation. -10.5 + 10.5 + x = -1.118033989 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + x = -1.118033989 + 10.5 x = -1.118033989 + 10.5 Combine like terms: -1.118033989 + 10.5 = 9.381966011 x = 9.381966011 Simplifying x = 9.381966011

Solution

The solution to the problem is based on the solutions from the subproblems. x = {11.618033989, 9.381966011}

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