4n^2+7n+1=0

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Solution for 4n^2+7n+1=0 equation:


Simplifying
4n2 + 7n + 1 = 0

Reorder the terms:
1 + 7n + 4n2 = 0

Solving
1 + 7n + 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 + 1.75n + n2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.75n + -0.25 + n2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.75n + n2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.75n + n2 = 0 + -0.25
1.75n + n2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.75n + n2 = -0.25

The n term is 1.75n.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75n + 0.765625 + n2 = -0.25 + 0.765625

Reorder the terms:
0.765625 + 1.75n + n2 = -0.25 + 0.765625

Combine like terms: -0.25 + 0.765625 = 0.515625
0.765625 + 1.75n + n2 = 0.515625

Factor a perfect square on the left side:
(n + 0.875)(n + 0.875) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(n + 0.875) equal to 0.718070331 and -0.718070331.

Subproblem 1

n + 0.875 = 0.718070331 Simplifying n + 0.875 = 0.718070331 Reorder the terms: 0.875 + n = 0.718070331 Solving 0.875 + n = 0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = 0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = 0.718070331 + -0.875 n = 0.718070331 + -0.875 Combine like terms: 0.718070331 + -0.875 = -0.156929669 n = -0.156929669 Simplifying n = -0.156929669

Subproblem 2

n + 0.875 = -0.718070331 Simplifying n + 0.875 = -0.718070331 Reorder the terms: 0.875 + n = -0.718070331 Solving 0.875 + n = -0.718070331 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + n = -0.718070331 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + n = -0.718070331 + -0.875 n = -0.718070331 + -0.875 Combine like terms: -0.718070331 + -0.875 = -1.593070331 n = -1.593070331 Simplifying n = -1.593070331

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.156929669, -1.593070331}

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