2n^2-3n=1

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Solution for 2n^2-3n=1 equation:


Simplifying
2n2 + -3n = 1

Reorder the terms:
-3n + 2n2 = 1

Solving
-3n + 2n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + -3n + 2n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -3n + 2n2 = 0

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

Divide each side by '2'.
-0.5 + -1.5n + n2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + -1.5n + 0.5 + n2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + -1.5n + n2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + -1.5n + n2 = 0 + 0.5
-1.5n + n2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
-1.5n + n2 = 0.5

The n term is -1.5n.  Take half its coefficient (-0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
-1.5n + 0.5625 + n2 = 0.5 + 0.5625

Reorder the terms:
0.5625 + -1.5n + n2 = 0.5 + 0.5625

Combine like terms: 0.5 + 0.5625 = 1.0625
0.5625 + -1.5n + n2 = 1.0625

Factor a perfect square on the left side:
(n + -0.75)(n + -0.75) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(n + -0.75) equal to 1.030776406 and -1.030776406.

Subproblem 1

n + -0.75 = 1.030776406 Simplifying n + -0.75 = 1.030776406 Reorder the terms: -0.75 + n = 1.030776406 Solving -0.75 + n = 1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = 1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = 1.030776406 + 0.75 n = 1.030776406 + 0.75 Combine like terms: 1.030776406 + 0.75 = 1.780776406 n = 1.780776406 Simplifying n = 1.780776406

Subproblem 2

n + -0.75 = -1.030776406 Simplifying n + -0.75 = -1.030776406 Reorder the terms: -0.75 + n = -1.030776406 Solving -0.75 + n = -1.030776406 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + n = -1.030776406 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + n = -1.030776406 + 0.75 n = -1.030776406 + 0.75 Combine like terms: -1.030776406 + 0.75 = -0.280776406 n = -0.280776406 Simplifying n = -0.280776406

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.780776406, -0.280776406}

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