5n^2+12n=-3

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Solution for 5n^2+12n=-3 equation:


Simplifying
5n2 + 12n = -3

Reorder the terms:
12n + 5n2 = -3

Solving
12n + 5n2 = -3

Solving for variable 'n'.

Reorder the terms:
3 + 12n + 5n2 = -3 + 3

Combine like terms: -3 + 3 = 0
3 + 12n + 5n2 = 0

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

Divide each side by '5'.
0.6 + 2.4n + n2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 2.4n + -0.6 + n2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 2.4n + n2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 2.4n + n2 = 0 + -0.6
2.4n + n2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
2.4n + n2 = -0.6

The n term is 2.4n.  Take half its coefficient (1.2).
Square it (1.44) and add it to both sides.

Add '1.44' to each side of the equation.
2.4n + 1.44 + n2 = -0.6 + 1.44

Reorder the terms:
1.44 + 2.4n + n2 = -0.6 + 1.44

Combine like terms: -0.6 + 1.44 = 0.84
1.44 + 2.4n + n2 = 0.84

Factor a perfect square on the left side:
(n + 1.2)(n + 1.2) = 0.84

Calculate the square root of the right side: 0.916515139

Break this problem into two subproblems by setting 
(n + 1.2) equal to 0.916515139 and -0.916515139.

Subproblem 1

n + 1.2 = 0.916515139 Simplifying n + 1.2 = 0.916515139 Reorder the terms: 1.2 + n = 0.916515139 Solving 1.2 + n = 0.916515139 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.2' to each side of the equation. 1.2 + -1.2 + n = 0.916515139 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + n = 0.916515139 + -1.2 n = 0.916515139 + -1.2 Combine like terms: 0.916515139 + -1.2 = -0.283484861 n = -0.283484861 Simplifying n = -0.283484861

Subproblem 2

n + 1.2 = -0.916515139 Simplifying n + 1.2 = -0.916515139 Reorder the terms: 1.2 + n = -0.916515139 Solving 1.2 + n = -0.916515139 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-1.2' to each side of the equation. 1.2 + -1.2 + n = -0.916515139 + -1.2 Combine like terms: 1.2 + -1.2 = 0.0 0.0 + n = -0.916515139 + -1.2 n = -0.916515139 + -1.2 Combine like terms: -0.916515139 + -1.2 = -2.116515139 n = -2.116515139 Simplifying n = -2.116515139

Solution

The solution to the problem is based on the solutions from the subproblems. n = {-0.283484861, -2.116515139}

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