4n^2+n-1900=0

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Solution for 4n^2+n-1900=0 equation:


Simplifying
4n2 + n + -1900 = 0

Reorder the terms:
-1900 + n + 4n2 = 0

Solving
-1900 + n + 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'.
-475 + 0.25n + n2 = 0

Move the constant term to the right:

Add '475' to each side of the equation.
-475 + 0.25n + 475 + n2 = 0 + 475

Reorder the terms:
-475 + 475 + 0.25n + n2 = 0 + 475

Combine like terms: -475 + 475 = 0
0 + 0.25n + n2 = 0 + 475
0.25n + n2 = 0 + 475

Combine like terms: 0 + 475 = 475
0.25n + n2 = 475

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.25n + 0.25 + n2 = 475 + 0.25

Reorder the terms:
0.25 + 0.25n + n2 = 475 + 0.25

Combine like terms: 475 + 0.25 = 475.25
0.25 + 0.25n + n2 = 475.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 475.25

Calculate the square root of the right side: 21.800229357

Break this problem into two subproblems by setting 
(n + 0.5) equal to 21.800229357 and -21.800229357.

Subproblem 1

n + 0.5 = 21.800229357 Simplifying n + 0.5 = 21.800229357 Reorder the terms: 0.5 + n = 21.800229357 Solving 0.5 + n = 21.800229357 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 21.800229357 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 21.800229357 + -0.5 n = 21.800229357 + -0.5 Combine like terms: 21.800229357 + -0.5 = 21.300229357 n = 21.300229357 Simplifying n = 21.300229357

Subproblem 2

n + 0.5 = -21.800229357 Simplifying n + 0.5 = -21.800229357 Reorder the terms: 0.5 + n = -21.800229357 Solving 0.5 + n = -21.800229357 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -21.800229357 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -21.800229357 + -0.5 n = -21.800229357 + -0.5 Combine like terms: -21.800229357 + -0.5 = -22.300229357 n = -22.300229357 Simplifying n = -22.300229357

Solution

The solution to the problem is based on the solutions from the subproblems. n = {21.300229357, -22.300229357}

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