2n^2-5n=1

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Solution for 2n^2-5n=1 equation:


Simplifying
2n2 + -5n = 1

Reorder the terms:
-5n + 2n2 = 1

Solving
-5n + 2n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -5n + 2n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -5n + 2n2 = 0

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

Divide each side by '2'.
-0.5 + -2.5n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -2.5n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -2.5n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -2.5n + n2 = 0 + 0.5
-2.5n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-2.5n + n2 = 0.5

The n term is -2.5n.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5n + 1.5625 + n2 = 0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5n + n2 = 0.5 + 1.5625

Combine like terms: 0.5 + 1.5625 = 2.0625
1.5625 + -2.5n + n2 = 2.0625

Factor a perfect square on the left side:
(n + -1.25)(n + -1.25) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(n + -1.25) equal to 1.436140662 and -1.436140662.

Subproblem 1

n + -1.25 = 1.436140662 Simplifying n + -1.25 = 1.436140662 Reorder the terms: -1.25 + n = 1.436140662 Solving -1.25 + n = 1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = 1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = 1.436140662 + 1.25 n = 1.436140662 + 1.25 Combine like terms: 1.436140662 + 1.25 = 2.686140662 n = 2.686140662 Simplifying n = 2.686140662

Subproblem 2

n + -1.25 = -1.436140662 Simplifying n + -1.25 = -1.436140662 Reorder the terms: -1.25 + n = -1.436140662 Solving -1.25 + n = -1.436140662 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + n = -1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + n = -1.436140662 + 1.25 n = -1.436140662 + 1.25 Combine like terms: -1.436140662 + 1.25 = -0.186140662 n = -0.186140662 Simplifying n = -0.186140662

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.686140662, -0.186140662}

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