4x^2-28x+41=0

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Solution for 4x^2-28x+41=0 equation:


Simplifying
4x2 + -28x + 41 = 0

Reorder the terms:
41 + -28x + 4x2 = 0

Solving
41 + -28x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
10.25 + -7x + x2 = 0

Move the constant term to the right:

Add '-10.25' to each side of the equation.
10.25 + -7x + -10.25 + x2 = 0 + -10.25

Reorder the terms:
10.25 + -10.25 + -7x + x2 = 0 + -10.25

Combine like terms: 10.25 + -10.25 = 0.00
0.00 + -7x + x2 = 0 + -10.25
-7x + x2 = 0 + -10.25

Combine like terms: 0 + -10.25 = -10.25
-7x + x2 = -10.25

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = -10.25 + 12.25

Reorder the terms:
12.25 + -7x + x2 = -10.25 + 12.25

Combine like terms: -10.25 + 12.25 = 2
12.25 + -7x + x2 = 2

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + -3.5) equal to 1.414213562 and -1.414213562.

Subproblem 1

x + -3.5 = 1.414213562 Simplifying x + -3.5 = 1.414213562 Reorder the terms: -3.5 + x = 1.414213562 Solving -3.5 + x = 1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = 1.414213562 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = 1.414213562 + 3.5 x = 1.414213562 + 3.5 Combine like terms: 1.414213562 + 3.5 = 4.914213562 x = 4.914213562 Simplifying x = 4.914213562

Subproblem 2

x + -3.5 = -1.414213562 Simplifying x + -3.5 = -1.414213562 Reorder the terms: -3.5 + x = -1.414213562 Solving -3.5 + x = -1.414213562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + x = -1.414213562 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + x = -1.414213562 + 3.5 x = -1.414213562 + 3.5 Combine like terms: -1.414213562 + 3.5 = 2.085786438 x = 2.085786438 Simplifying x = 2.085786438

Solution

The solution to the problem is based on the solutions from the subproblems. x = {4.914213562, 2.085786438}

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