4n^2+14n=1

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Solution for 4n^2+14n=1 equation:


Simplifying
4n2 + 14n = 1

Reorder the terms:
14n + 4n2 = 1

Solving
14n + 4n2 = 1

Solving for variable 'n'.

Reorder the terms:
-1 + 14n + 4n2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + 14n + 4n2 = 0

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

Divide each side by '4'.
-0.25 + 3.5n + n2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 3.5n + 0.25 + n2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 3.5n + n2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 3.5n + n2 = 0 + 0.25
3.5n + n2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
3.5n + n2 = 0.25

The n term is 3.5n.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5n + 3.0625 + n2 = 0.25 + 3.0625

Reorder the terms:
3.0625 + 3.5n + n2 = 0.25 + 3.0625

Combine like terms: 0.25 + 3.0625 = 3.3125
3.0625 + 3.5n + n2 = 3.3125

Factor a perfect square on the left side:
(n + 1.75)(n + 1.75) = 3.3125

Calculate the square root of the right side: 1.820027472

Break this problem into two subproblems by setting 
(n + 1.75) equal to 1.820027472 and -1.820027472.

Subproblem 1

n + 1.75 = 1.820027472 Simplifying n + 1.75 = 1.820027472 Reorder the terms: 1.75 + n = 1.820027472 Solving 1.75 + n = 1.820027472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = 1.820027472 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = 1.820027472 + -1.75 n = 1.820027472 + -1.75 Combine like terms: 1.820027472 + -1.75 = 0.070027472 n = 0.070027472 Simplifying n = 0.070027472

Subproblem 2

n + 1.75 = -1.820027472 Simplifying n + 1.75 = -1.820027472 Reorder the terms: 1.75 + n = -1.820027472 Solving 1.75 + n = -1.820027472 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + n = -1.820027472 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + n = -1.820027472 + -1.75 n = -1.820027472 + -1.75 Combine like terms: -1.820027472 + -1.75 = -3.570027472 n = -3.570027472 Simplifying n = -3.570027472

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.070027472, -3.570027472}

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