7n^2+22n+5=0

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Solution for 7n^2+22n+5=0 equation:


Simplifying
7n2 + 22n + 5 = 0

Reorder the terms:
5 + 22n + 7n2 = 0

Solving
5 + 22n + 7n2 = 0

Solving for variable 'n'.

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

Divide each side by '7'.
0.7142857143 + 3.142857143n + n2 = 0

Move the constant term to the right:

Add '-0.7142857143' to each side of the equation.
0.7142857143 + 3.142857143n + -0.7142857143 + n2 = 0 + -0.7142857143

Reorder the terms:
0.7142857143 + -0.7142857143 + 3.142857143n + n2 = 0 + -0.7142857143

Combine like terms: 0.7142857143 + -0.7142857143 = 0.0000000000
0.0000000000 + 3.142857143n + n2 = 0 + -0.7142857143
3.142857143n + n2 = 0 + -0.7142857143

Combine like terms: 0 + -0.7142857143 = -0.7142857143
3.142857143n + n2 = -0.7142857143

The n term is 3.142857143n.  Take half its coefficient (1.571428572).
Square it (2.469387757) and add it to both sides.

Add '2.469387757' to each side of the equation.
3.142857143n + 2.469387757 + n2 = -0.7142857143 + 2.469387757

Reorder the terms:
2.469387757 + 3.142857143n + n2 = -0.7142857143 + 2.469387757

Combine like terms: -0.7142857143 + 2.469387757 = 1.7551020427
2.469387757 + 3.142857143n + n2 = 1.7551020427

Factor a perfect square on the left side:
(n + 1.571428572)(n + 1.571428572) = 1.7551020427

Calculate the square root of the right side: 1.324802643

Break this problem into two subproblems by setting 
(n + 1.571428572) equal to 1.324802643 and -1.324802643.

Subproblem 1

n + 1.571428572 = 1.324802643 Simplifying n + 1.571428572 = 1.324802643 Reorder the terms: 1.571428572 + n = 1.324802643 Solving 1.571428572 + n = 1.324802643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + n = 1.324802643 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + n = 1.324802643 + -1.571428572 n = 1.324802643 + -1.571428572 Combine like terms: 1.324802643 + -1.571428572 = -0.246625929 n = -0.246625929 Simplifying n = -0.246625929

Subproblem 2

n + 1.571428572 = -1.324802643 Simplifying n + 1.571428572 = -1.324802643 Reorder the terms: 1.571428572 + n = -1.324802643 Solving 1.571428572 + n = -1.324802643 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.571428572' to each side of the equation. 1.571428572 + -1.571428572 + n = -1.324802643 + -1.571428572 Combine like terms: 1.571428572 + -1.571428572 = 0.000000000 0.000000000 + n = -1.324802643 + -1.571428572 n = -1.324802643 + -1.571428572 Combine like terms: -1.324802643 + -1.571428572 = -2.896231215 n = -2.896231215 Simplifying n = -2.896231215

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.246625929, -2.896231215}

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