5n^2-30n+16=0

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Solution for 5n^2-30n+16=0 equation:


Simplifying
5n2 + -30n + 16 = 0

Reorder the terms:
16 + -30n + 5n2 = 0

Solving
16 + -30n + 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 + -6n + n2 = 0

Move the constant term to the right:

Add '-3.2' to each side of the equation.
3.2 + -6n + -3.2 + n2 = 0 + -3.2

Reorder the terms:
3.2 + -3.2 + -6n + n2 = 0 + -3.2

Combine like terms: 3.2 + -3.2 = 0.0
0.0 + -6n + n2 = 0 + -3.2
-6n + n2 = 0 + -3.2

Combine like terms: 0 + -3.2 = -3.2
-6n + n2 = -3.2

The n term is -6n.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6n + 9 + n2 = -3.2 + 9

Reorder the terms:
9 + -6n + n2 = -3.2 + 9

Combine like terms: -3.2 + 9 = 5.8
9 + -6n + n2 = 5.8

Factor a perfect square on the left side:
(n + -3)(n + -3) = 5.8

Calculate the square root of the right side: 2.408318916

Break this problem into two subproblems by setting 
(n + -3) equal to 2.408318916 and -2.408318916.

Subproblem 1

n + -3 = 2.408318916 Simplifying n + -3 = 2.408318916 Reorder the terms: -3 + n = 2.408318916 Solving -3 + n = 2.408318916 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = 2.408318916 + 3 Combine like terms: -3 + 3 = 0 0 + n = 2.408318916 + 3 n = 2.408318916 + 3 Combine like terms: 2.408318916 + 3 = 5.408318916 n = 5.408318916 Simplifying n = 5.408318916

Subproblem 2

n + -3 = -2.408318916 Simplifying n + -3 = -2.408318916 Reorder the terms: -3 + n = -2.408318916 Solving -3 + n = -2.408318916 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + n = -2.408318916 + 3 Combine like terms: -3 + 3 = 0 0 + n = -2.408318916 + 3 n = -2.408318916 + 3 Combine like terms: -2.408318916 + 3 = 0.591681084 n = 0.591681084 Simplifying n = 0.591681084

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.408318916, 0.591681084}

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