4n^2+5n-1=0

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Solution for 4n^2+5n-1=0 equation:


Simplifying
4n2 + 5n + -1 = 0

Reorder the terms:
-1 + 5n + 4n2 = 0

Solving
-1 + 5n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-0.25 + 1.25n + n2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 1.25n + 0.25 + n2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 1.25n + n2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 1.25n + n2 = 0 + 0.25
1.25n + n2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
1.25n + n2 = 0.25

The n term is 1.25n.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25n + 0.390625 + n2 = 0.25 + 0.390625

Reorder the terms:
0.390625 + 1.25n + n2 = 0.25 + 0.390625

Combine like terms: 0.25 + 0.390625 = 0.640625
0.390625 + 1.25n + n2 = 0.640625

Factor a perfect square on the left side:
(n + 0.625)(n + 0.625) = 0.640625

Calculate the square root of the right side: 0.80039053

Break this problem into two subproblems by setting 
(n + 0.625) equal to 0.80039053 and -0.80039053.

Subproblem 1

n + 0.625 = 0.80039053 Simplifying n + 0.625 = 0.80039053 Reorder the terms: 0.625 + n = 0.80039053 Solving 0.625 + n = 0.80039053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = 0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = 0.80039053 + -0.625 n = 0.80039053 + -0.625 Combine like terms: 0.80039053 + -0.625 = 0.17539053 n = 0.17539053 Simplifying n = 0.17539053

Subproblem 2

n + 0.625 = -0.80039053 Simplifying n + 0.625 = -0.80039053 Reorder the terms: 0.625 + n = -0.80039053 Solving 0.625 + n = -0.80039053 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = -0.80039053 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = -0.80039053 + -0.625 n = -0.80039053 + -0.625 Combine like terms: -0.80039053 + -0.625 = -1.42539053 n = -1.42539053 Simplifying n = -1.42539053

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.17539053, -1.42539053}

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