14x^2-41x+5=0

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Solution for 14x^2-41x+5=0 equation:


Simplifying
14x2 + -41x + 5 = 0

Reorder the terms:
5 + -41x + 14x2 = 0

Solving
5 + -41x + 14x2 = 0

Solving for variable 'x'.

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

Divide each side by '14'.
0.3571428571 + -2.928571429x + x2 = 0

Move the constant term to the right:

Add '-0.3571428571' to each side of the equation.
0.3571428571 + -2.928571429x + -0.3571428571 + x2 = 0 + -0.3571428571

Reorder the terms:
0.3571428571 + -0.3571428571 + -2.928571429x + x2 = 0 + -0.3571428571

Combine like terms: 0.3571428571 + -0.3571428571 = 0.0000000000
0.0000000000 + -2.928571429x + x2 = 0 + -0.3571428571
-2.928571429x + x2 = 0 + -0.3571428571

Combine like terms: 0 + -0.3571428571 = -0.3571428571
-2.928571429x + x2 = -0.3571428571

The x term is -2.928571429x.  Take half its coefficient (-1.464285715).
Square it (2.144132655) and add it to both sides.

Add '2.144132655' to each side of the equation.
-2.928571429x + 2.144132655 + x2 = -0.3571428571 + 2.144132655

Reorder the terms:
2.144132655 + -2.928571429x + x2 = -0.3571428571 + 2.144132655

Combine like terms: -0.3571428571 + 2.144132655 = 1.7869897979
2.144132655 + -2.928571429x + x2 = 1.7869897979

Factor a perfect square on the left side:
(x + -1.464285715)(x + -1.464285715) = 1.7869897979

Calculate the square root of the right side: 1.336783377

Break this problem into two subproblems by setting 
(x + -1.464285715) equal to 1.336783377 and -1.336783377.

Subproblem 1

x + -1.464285715 = 1.336783377 Simplifying x + -1.464285715 = 1.336783377 Reorder the terms: -1.464285715 + x = 1.336783377 Solving -1.464285715 + x = 1.336783377 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.464285715' to each side of the equation. -1.464285715 + 1.464285715 + x = 1.336783377 + 1.464285715 Combine like terms: -1.464285715 + 1.464285715 = 0.000000000 0.000000000 + x = 1.336783377 + 1.464285715 x = 1.336783377 + 1.464285715 Combine like terms: 1.336783377 + 1.464285715 = 2.801069092 x = 2.801069092 Simplifying x = 2.801069092

Subproblem 2

x + -1.464285715 = -1.336783377 Simplifying x + -1.464285715 = -1.336783377 Reorder the terms: -1.464285715 + x = -1.336783377 Solving -1.464285715 + x = -1.336783377 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.464285715' to each side of the equation. -1.464285715 + 1.464285715 + x = -1.336783377 + 1.464285715 Combine like terms: -1.464285715 + 1.464285715 = 0.000000000 0.000000000 + x = -1.336783377 + 1.464285715 x = -1.336783377 + 1.464285715 Combine like terms: -1.336783377 + 1.464285715 = 0.127502338 x = 0.127502338 Simplifying x = 0.127502338

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2.801069092, 0.127502338}

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