4n^2-n-6=0

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Solution for 4n^2-n-6=0 equation:


Simplifying
4n2 + -1n + -6 = 0

Reorder the terms:
-6 + -1n + 4n2 = 0

Solving
-6 + -1n + 4n2 = 0

Solving for variable 'n'.

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

Divide each side by '4'.
-1.5 + -0.25n + n2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -0.25n + 1.5 + n2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -0.25n + n2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -0.25n + n2 = 0 + 1.5
-0.25n + n2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-0.25n + n2 = 1.5

The n term is -0.25n.  Take half its coefficient (-0.125).
Square it (0.015625) and add it to both sides.

Add '0.015625' to each side of the equation.
-0.25n + 0.015625 + n2 = 1.5 + 0.015625

Reorder the terms:
0.015625 + -0.25n + n2 = 1.5 + 0.015625

Combine like terms: 1.5 + 0.015625 = 1.515625
0.015625 + -0.25n + n2 = 1.515625

Factor a perfect square on the left side:
(n + -0.125)(n + -0.125) = 1.515625

Calculate the square root of the right side: 1.231107225

Break this problem into two subproblems by setting 
(n + -0.125) equal to 1.231107225 and -1.231107225.

Subproblem 1

n + -0.125 = 1.231107225 Simplifying n + -0.125 = 1.231107225 Reorder the terms: -0.125 + n = 1.231107225 Solving -0.125 + n = 1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = 1.231107225 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = 1.231107225 + 0.125 n = 1.231107225 + 0.125 Combine like terms: 1.231107225 + 0.125 = 1.356107225 n = 1.356107225 Simplifying n = 1.356107225

Subproblem 2

n + -0.125 = -1.231107225 Simplifying n + -0.125 = -1.231107225 Reorder the terms: -0.125 + n = -1.231107225 Solving -0.125 + n = -1.231107225 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + n = -1.231107225 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + n = -1.231107225 + 0.125 n = -1.231107225 + 0.125 Combine like terms: -1.231107225 + 0.125 = -1.106107225 n = -1.106107225 Simplifying n = -1.106107225

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.356107225, -1.106107225}

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