5n^2-20n+6=10

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Solution for 5n^2-20n+6=10 equation:


Simplifying
5n2 + -20n + 6 = 10

Reorder the terms:
6 + -20n + 5n2 = 10

Solving
6 + -20n + 5n2 = 10

Solving for variable 'n'.

Reorder the terms:
6 + -10 + -20n + 5n2 = 10 + -10

Combine like terms: 6 + -10 = -4
-4 + -20n + 5n2 = 10 + -10

Combine like terms: 10 + -10 = 0
-4 + -20n + 5n2 = 0

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

Divide each side by '5'.
-0.8 + -4n + n2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + -4n + 0.8 + n2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + -4n + n2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + -4n + n2 = 0 + 0.8
-4n + n2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
-4n + n2 = 0.8

The n term is -4n.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4n + 4 + n2 = 0.8 + 4

Reorder the terms:
4 + -4n + n2 = 0.8 + 4

Combine like terms: 0.8 + 4 = 4.8
4 + -4n + n2 = 4.8

Factor a perfect square on the left side:
(n + -2)(n + -2) = 4.8

Calculate the square root of the right side: 2.19089023

Break this problem into two subproblems by setting 
(n + -2) equal to 2.19089023 and -2.19089023.

Subproblem 1

n + -2 = 2.19089023 Simplifying n + -2 = 2.19089023 Reorder the terms: -2 + n = 2.19089023 Solving -2 + n = 2.19089023 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = 2.19089023 + 2 Combine like terms: -2 + 2 = 0 0 + n = 2.19089023 + 2 n = 2.19089023 + 2 Combine like terms: 2.19089023 + 2 = 4.19089023 n = 4.19089023 Simplifying n = 4.19089023

Subproblem 2

n + -2 = -2.19089023 Simplifying n + -2 = -2.19089023 Reorder the terms: -2 + n = -2.19089023 Solving -2 + n = -2.19089023 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + n = -2.19089023 + 2 Combine like terms: -2 + 2 = 0 0 + n = -2.19089023 + 2 n = -2.19089023 + 2 Combine like terms: -2.19089023 + 2 = -0.19089023 n = -0.19089023 Simplifying n = -0.19089023

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.19089023, -0.19089023}

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