2n^2+5n-4=0

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Solution for 2n^2+5n-4=0 equation:


Simplifying
2n2 + 5n + -4 = 0

Reorder the terms:
-4 + 5n + 2n2 = 0

Solving
-4 + 5n + 2n2 = 0

Solving for variable 'n'.

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

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

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 2.5n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 2.5n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 2.5n + n2 = 0 + 2
2.5n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
2.5n + n2 = 2

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 = 2 + 1.5625

Reorder the terms:
1.5625 + 2.5n + n2 = 2 + 1.5625

Combine like terms: 2 + 1.5625 = 3.5625
1.5625 + 2.5n + n2 = 3.5625

Factor a perfect square on the left side:
(n + 1.25)(n + 1.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(n + 1.25) equal to 1.887458609 and -1.887458609.

Subproblem 1

n + 1.25 = 1.887458609 Simplifying n + 1.25 = 1.887458609 Reorder the terms: 1.25 + n = 1.887458609 Solving 1.25 + n = 1.887458609 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 = 1.887458609 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = 1.887458609 + -1.25 n = 1.887458609 + -1.25 Combine like terms: 1.887458609 + -1.25 = 0.637458609 n = 0.637458609 Simplifying n = 0.637458609

Subproblem 2

n + 1.25 = -1.887458609 Simplifying n + 1.25 = -1.887458609 Reorder the terms: 1.25 + n = -1.887458609 Solving 1.25 + n = -1.887458609 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 = -1.887458609 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + n = -1.887458609 + -1.25 n = -1.887458609 + -1.25 Combine like terms: -1.887458609 + -1.25 = -3.137458609 n = -3.137458609 Simplifying n = -3.137458609

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.637458609, -3.137458609}

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