5n^2+32n=-21

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Solution for 5n^2+32n=-21 equation:


Simplifying
5n2 + 32n = -21

Reorder the terms:
32n + 5n2 = -21

Solving
32n + 5n2 = -21

Solving for variable 'n'.

Reorder the terms:
21 + 32n + 5n2 = -21 + 21

Combine like terms: -21 + 21 = 0
21 + 32n + 5n2 = 0

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

Divide each side by '5'.
4.2 + 6.4n + n2 = 0

Move the constant term to the right:

Add '-4.2' to each side of the equation.
4.2 + 6.4n + -4.2 + n2 = 0 + -4.2

Reorder the terms:
4.2 + -4.2 + 6.4n + n2 = 0 + -4.2

Combine like terms: 4.2 + -4.2 = 0.0
0.0 + 6.4n + n2 = 0 + -4.2
6.4n + n2 = 0 + -4.2

Combine like terms: 0 + -4.2 = -4.2
6.4n + n2 = -4.2

The n term is 6.4n.  Take half its coefficient (3.2).
Square it (10.24) and add it to both sides.

Add '10.24' to each side of the equation.
6.4n + 10.24 + n2 = -4.2 + 10.24

Reorder the terms:
10.24 + 6.4n + n2 = -4.2 + 10.24

Combine like terms: -4.2 + 10.24 = 6.04
10.24 + 6.4n + n2 = 6.04

Factor a perfect square on the left side:
(n + 3.2)(n + 3.2) = 6.04

Calculate the square root of the right side: 2.457641145

Break this problem into two subproblems by setting 
(n + 3.2) equal to 2.457641145 and -2.457641145.

Subproblem 1

n + 3.2 = 2.457641145 Simplifying n + 3.2 = 2.457641145 Reorder the terms: 3.2 + n = 2.457641145 Solving 3.2 + n = 2.457641145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + n = 2.457641145 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + n = 2.457641145 + -3.2 n = 2.457641145 + -3.2 Combine like terms: 2.457641145 + -3.2 = -0.742358855 n = -0.742358855 Simplifying n = -0.742358855

Subproblem 2

n + 3.2 = -2.457641145 Simplifying n + 3.2 = -2.457641145 Reorder the terms: 3.2 + n = -2.457641145 Solving 3.2 + n = -2.457641145 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-3.2' to each side of the equation. 3.2 + -3.2 + n = -2.457641145 + -3.2 Combine like terms: 3.2 + -3.2 = 0.0 0.0 + n = -2.457641145 + -3.2 n = -2.457641145 + -3.2 Combine like terms: -2.457641145 + -3.2 = -5.657641145 n = -5.657641145 Simplifying n = -5.657641145

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.742358855, -5.657641145}

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