5n^2-10n+4=0

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Solution for 5n^2-10n+4=0 equation:


Simplifying
5n2 + -10n + 4 = 0

Reorder the terms:
4 + -10n + 5n2 = 0

Solving
4 + -10n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
0.8 + -2n + n2 = 0

Move the constant term to the right:

Add '-0.8' to each side of the equation.
0.8 + -2n + -0.8 + n2 = 0 + -0.8

Reorder the terms:
0.8 + -0.8 + -2n + n2 = 0 + -0.8

Combine like terms: 0.8 + -0.8 = 0.0
0.0 + -2n + n2 = 0 + -0.8
-2n + n2 = 0 + -0.8

Combine like terms: 0 + -0.8 = -0.8
-2n + n2 = -0.8

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = -0.8 + 1

Reorder the terms:
1 + -2n + n2 = -0.8 + 1

Combine like terms: -0.8 + 1 = 0.2
1 + -2n + n2 = 0.2

Factor a perfect square on the left side:
(n + -1)(n + -1) = 0.2

Calculate the square root of the right side: 0.447213596

Break this problem into two subproblems by setting 
(n + -1) equal to 0.447213596 and -0.447213596.

Subproblem 1

n + -1 = 0.447213596 Simplifying n + -1 = 0.447213596 Reorder the terms: -1 + n = 0.447213596 Solving -1 + n = 0.447213596 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 0.447213596 + 1 Combine like terms: -1 + 1 = 0 0 + n = 0.447213596 + 1 n = 0.447213596 + 1 Combine like terms: 0.447213596 + 1 = 1.447213596 n = 1.447213596 Simplifying n = 1.447213596

Subproblem 2

n + -1 = -0.447213596 Simplifying n + -1 = -0.447213596 Reorder the terms: -1 + n = -0.447213596 Solving -1 + n = -0.447213596 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -0.447213596 + 1 Combine like terms: -1 + 1 = 0 0 + n = -0.447213596 + 1 n = -0.447213596 + 1 Combine like terms: -0.447213596 + 1 = 0.552786404 n = 0.552786404 Simplifying n = 0.552786404

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.447213596, 0.552786404}

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