5n^2+21n+20=0

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Solution for 5n^2+21n+20=0 equation:


Simplifying
5n2 + 21n + 20 = 0

Reorder the terms:
20 + 21n + 5n2 = 0

Solving
20 + 21n + 5n2 = 0

Solving for variable 'n'.

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

Divide each side by '5'.
4 + 4.2n + n2 = 0

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 4.2n + -4 + n2 = 0 + -4

Reorder the terms:
4 + -4 + 4.2n + n2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 4.2n + n2 = 0 + -4
4.2n + n2 = 0 + -4

Combine like terms: 0 + -4 = -4
4.2n + n2 = -4

The n term is 4.2n.  Take half its coefficient (2.1).
Square it (4.41) and add it to both sides.

Add '4.41' to each side of the equation.
4.2n + 4.41 + n2 = -4 + 4.41

Reorder the terms:
4.41 + 4.2n + n2 = -4 + 4.41

Combine like terms: -4 + 4.41 = 0.41
4.41 + 4.2n + n2 = 0.41

Factor a perfect square on the left side:
(n + 2.1)(n + 2.1) = 0.41

Calculate the square root of the right side: 0.640312424

Break this problem into two subproblems by setting 
(n + 2.1) equal to 0.640312424 and -0.640312424.

Subproblem 1

n + 2.1 = 0.640312424 Simplifying n + 2.1 = 0.640312424 Reorder the terms: 2.1 + n = 0.640312424 Solving 2.1 + n = 0.640312424 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + n = 0.640312424 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + n = 0.640312424 + -2.1 n = 0.640312424 + -2.1 Combine like terms: 0.640312424 + -2.1 = -1.459687576 n = -1.459687576 Simplifying n = -1.459687576

Subproblem 2

n + 2.1 = -0.640312424 Simplifying n + 2.1 = -0.640312424 Reorder the terms: 2.1 + n = -0.640312424 Solving 2.1 + n = -0.640312424 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2.1' to each side of the equation. 2.1 + -2.1 + n = -0.640312424 + -2.1 Combine like terms: 2.1 + -2.1 = 0.0 0.0 + n = -0.640312424 + -2.1 n = -0.640312424 + -2.1 Combine like terms: -0.640312424 + -2.1 = -2.740312424 n = -2.740312424 Simplifying n = -2.740312424

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-1.459687576, -2.740312424}

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