5n^2-2n-15=0

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Solution for 5n^2-2n-15=0 equation:


Simplifying
5n2 + -2n + -15 = 0

Reorder the terms:
-15 + -2n + 5n2 = 0

Solving
-15 + -2n + 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'.
-3 + -0.4n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.4n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.4n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.4n + n2 = 0 + 3
-0.4n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.4n + n2 = 3

The n term is -0.4n.  Take half its coefficient (-0.2).
Square it (0.04) and add it to both sides.

Add '0.04' to each side of the equation.
-0.4n + 0.04 + n2 = 3 + 0.04

Reorder the terms:
0.04 + -0.4n + n2 = 3 + 0.04

Combine like terms: 3 + 0.04 = 3.04
0.04 + -0.4n + n2 = 3.04

Factor a perfect square on the left side:
(n + -0.2)(n + -0.2) = 3.04

Calculate the square root of the right side: 1.743559577

Break this problem into two subproblems by setting 
(n + -0.2) equal to 1.743559577 and -1.743559577.

Subproblem 1

n + -0.2 = 1.743559577 Simplifying n + -0.2 = 1.743559577 Reorder the terms: -0.2 + n = 1.743559577 Solving -0.2 + n = 1.743559577 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = 1.743559577 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = 1.743559577 + 0.2 n = 1.743559577 + 0.2 Combine like terms: 1.743559577 + 0.2 = 1.943559577 n = 1.943559577 Simplifying n = 1.943559577

Subproblem 2

n + -0.2 = -1.743559577 Simplifying n + -0.2 = -1.743559577 Reorder the terms: -0.2 + n = -1.743559577 Solving -0.2 + n = -1.743559577 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + n = -1.743559577 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + n = -1.743559577 + 0.2 n = -1.743559577 + 0.2 Combine like terms: -1.743559577 + 0.2 = -1.543559577 n = -1.543559577 Simplifying n = -1.543559577

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.943559577, -1.543559577}

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