10x^2-41x+32=0

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Solution for 10x^2-41x+32=0 equation:


Simplifying
10x2 + -41x + 32 = 0

Reorder the terms:
32 + -41x + 10x2 = 0

Solving
32 + -41x + 10x2 = 0

Solving for variable 'x'.

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

Divide each side by '10'.
3.2 + -4.1x + x2 = 0

Move the constant term to the right:

Add '-3.2' to each side of the equation.
3.2 + -4.1x + -3.2 + x2 = 0 + -3.2

Reorder the terms:
3.2 + -3.2 + -4.1x + x2 = 0 + -3.2

Combine like terms: 3.2 + -3.2 = 0.0
0.0 + -4.1x + x2 = 0 + -3.2
-4.1x + x2 = 0 + -3.2

Combine like terms: 0 + -3.2 = -3.2
-4.1x + x2 = -3.2

The x term is -4.1x.  Take half its coefficient (-2.05).
Square it (4.2025) and add it to both sides.

Add '4.2025' to each side of the equation.
-4.1x + 4.2025 + x2 = -3.2 + 4.2025

Reorder the terms:
4.2025 + -4.1x + x2 = -3.2 + 4.2025

Combine like terms: -3.2 + 4.2025 = 1.0025
4.2025 + -4.1x + x2 = 1.0025

Factor a perfect square on the left side:
(x + -2.05)(x + -2.05) = 1.0025

Calculate the square root of the right side: 1.00124922

Break this problem into two subproblems by setting 
(x + -2.05) equal to 1.00124922 and -1.00124922.

Subproblem 1

x + -2.05 = 1.00124922 Simplifying x + -2.05 = 1.00124922 Reorder the terms: -2.05 + x = 1.00124922 Solving -2.05 + x = 1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + x = 1.00124922 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + x = 1.00124922 + 2.05 x = 1.00124922 + 2.05 Combine like terms: 1.00124922 + 2.05 = 3.05124922 x = 3.05124922 Simplifying x = 3.05124922

Subproblem 2

x + -2.05 = -1.00124922 Simplifying x + -2.05 = -1.00124922 Reorder the terms: -2.05 + x = -1.00124922 Solving -2.05 + x = -1.00124922 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + x = -1.00124922 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + x = -1.00124922 + 2.05 x = -1.00124922 + 2.05 Combine like terms: -1.00124922 + 2.05 = 1.04875078 x = 1.04875078 Simplifying x = 1.04875078

Solution

The solution to the problem is based on the solutions from the subproblems. x = {3.05124922, 1.04875078}

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