5n^2-32n+16=0

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Solution for 5n^2-32n+16=0 equation:


Simplifying
5n2 + -32n + 16 = 0

Reorder the terms:
16 + -32n + 5n2 = 0

Solving
16 + -32n + 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'.
3.2 + -6.4n + n2 = 0

Move the constant term to the right:

Add '-3.2' to each side of the equation.
3.2 + -6.4n + -3.2 + n2 = 0 + -3.2

Reorder the terms:
3.2 + -3.2 + -6.4n + n2 = 0 + -3.2

Combine like terms: 3.2 + -3.2 = 0.0
0.0 + -6.4n + n2 = 0 + -3.2
-6.4n + n2 = 0 + -3.2

Combine like terms: 0 + -3.2 = -3.2
-6.4n + n2 = -3.2

The n term is -6.4n.  Take half its coefficient (-3.2).
Square it (10.24) and add it to both sides.

Add '10.24' to each side of the equation.
-6.4n + 10.24 + n2 = -3.2 + 10.24

Reorder the terms:
10.24 + -6.4n + n2 = -3.2 + 10.24

Combine like terms: -3.2 + 10.24 = 7.04
10.24 + -6.4n + n2 = 7.04

Factor a perfect square on the left side:
(n + -3.2)(n + -3.2) = 7.04

Calculate the square root of the right side: 2.653299832

Break this problem into two subproblems by setting 
(n + -3.2) equal to 2.653299832 and -2.653299832.

Subproblem 1

n + -3.2 = 2.653299832 Simplifying n + -3.2 = 2.653299832 Reorder the terms: -3.2 + n = 2.653299832 Solving -3.2 + n = 2.653299832 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = 2.653299832 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = 2.653299832 + 3.2 n = 2.653299832 + 3.2 Combine like terms: 2.653299832 + 3.2 = 5.853299832 n = 5.853299832 Simplifying n = 5.853299832

Subproblem 2

n + -3.2 = -2.653299832 Simplifying n + -3.2 = -2.653299832 Reorder the terms: -3.2 + n = -2.653299832 Solving -3.2 + n = -2.653299832 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3.2' to each side of the equation. -3.2 + 3.2 + n = -2.653299832 + 3.2 Combine like terms: -3.2 + 3.2 = 0.0 0.0 + n = -2.653299832 + 3.2 n = -2.653299832 + 3.2 Combine like terms: -2.653299832 + 3.2 = 0.546700168 n = 0.546700168 Simplifying n = 0.546700168

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.853299832, 0.546700168}

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