5n^2+n-3=0

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Solution for 5n^2+n-3=0 equation:


Simplifying
5n2 + n + -3 = 0

Reorder the terms:
-3 + n + 5n2 = 0

Solving
-3 + n + 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'.
-0.6 + 0.2n + n2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + 0.2n + 0.6 + n2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + 0.2n + n2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + 0.2n + n2 = 0 + 0.6
0.2n + n2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
0.2n + n2 = 0.6

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.2n + 0.25 + n2 = 0.6 + 0.25

Reorder the terms:
0.25 + 0.2n + n2 = 0.6 + 0.25

Combine like terms: 0.6 + 0.25 = 0.85
0.25 + 0.2n + n2 = 0.85

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.85

Calculate the square root of the right side: 0.921954446

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.921954446 and -0.921954446.

Subproblem 1

n + 0.5 = 0.921954446 Simplifying n + 0.5 = 0.921954446 Reorder the terms: 0.5 + n = 0.921954446 Solving 0.5 + n = 0.921954446 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 0.921954446 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.921954446 + -0.5 n = 0.921954446 + -0.5 Combine like terms: 0.921954446 + -0.5 = 0.421954446 n = 0.421954446 Simplifying n = 0.421954446

Subproblem 2

n + 0.5 = -0.921954446 Simplifying n + 0.5 = -0.921954446 Reorder the terms: 0.5 + n = -0.921954446 Solving 0.5 + n = -0.921954446 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -0.921954446 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.921954446 + -0.5 n = -0.921954446 + -0.5 Combine like terms: -0.921954446 + -0.5 = -1.421954446 n = -1.421954446 Simplifying n = -1.421954446

Solution

The solution to the problem is based on the solutions from the subproblems. n = {0.421954446, -1.421954446}

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