5n^2-5=-3n

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


Simplifying
5n2 + -5 = -3n

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

Solving
-5 + 5n2 = -3n

Solving for variable 'n'.

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

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

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

Divide each side by '5'.
-1 + 0.6n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 0.6n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 0.6n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 0.6n + n2 = 0 + 1
0.6n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
0.6n + n2 = 1

The n term is 0.6n.  Take half its coefficient (0.3).
Square it (0.09) and add it to both sides.

Add '0.09' to each side of the equation.
0.6n + 0.09 + n2 = 1 + 0.09

Reorder the terms:
0.09 + 0.6n + n2 = 1 + 0.09

Combine like terms: 1 + 0.09 = 1.09
0.09 + 0.6n + n2 = 1.09

Factor a perfect square on the left side:
(n + 0.3)(n + 0.3) = 1.09

Calculate the square root of the right side: 1.044030651

Break this problem into two subproblems by setting 
(n + 0.3) equal to 1.044030651 and -1.044030651.

Subproblem 1

n + 0.3 = 1.044030651 Simplifying n + 0.3 = 1.044030651 Reorder the terms: 0.3 + n = 1.044030651 Solving 0.3 + n = 1.044030651 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = 1.044030651 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = 1.044030651 + -0.3 n = 1.044030651 + -0.3 Combine like terms: 1.044030651 + -0.3 = 0.744030651 n = 0.744030651 Simplifying n = 0.744030651

Subproblem 2

n + 0.3 = -1.044030651 Simplifying n + 0.3 = -1.044030651 Reorder the terms: 0.3 + n = -1.044030651 Solving 0.3 + n = -1.044030651 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + n = -1.044030651 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + n = -1.044030651 + -0.3 n = -1.044030651 + -0.3 Combine like terms: -1.044030651 + -0.3 = -1.344030651 n = -1.344030651 Simplifying n = -1.344030651

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.744030651, -1.344030651}

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