4n^2-21n-20=0

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Solution for 4n^2-21n-20=0 equation:


Simplifying
4n2 + -21n + -20 = 0

Reorder the terms:
-20 + -21n + 4n2 = 0

Solving
-20 + -21n + 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'.
-5 + -5.25n + n2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -5.25n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -5.25n + n2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -5.25n + n2 = 0 + 5
-5.25n + n2 = 0 + 5

Combine like terms: 0 + 5 = 5
-5.25n + n2 = 5

The n term is -5.25n.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25n + 6.890625 + n2 = 5 + 6.890625

Reorder the terms:
6.890625 + -5.25n + n2 = 5 + 6.890625

Combine like terms: 5 + 6.890625 = 11.890625
6.890625 + -5.25n + n2 = 11.890625

Factor a perfect square on the left side:
(n + -2.625)(n + -2.625) = 11.890625

Calculate the square root of the right side: 3.448278556

Break this problem into two subproblems by setting 
(n + -2.625) equal to 3.448278556 and -3.448278556.

Subproblem 1

n + -2.625 = 3.448278556 Simplifying n + -2.625 = 3.448278556 Reorder the terms: -2.625 + n = 3.448278556 Solving -2.625 + n = 3.448278556 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = 3.448278556 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = 3.448278556 + 2.625 n = 3.448278556 + 2.625 Combine like terms: 3.448278556 + 2.625 = 6.073278556 n = 6.073278556 Simplifying n = 6.073278556

Subproblem 2

n + -2.625 = -3.448278556 Simplifying n + -2.625 = -3.448278556 Reorder the terms: -2.625 + n = -3.448278556 Solving -2.625 + n = -3.448278556 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + n = -3.448278556 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + n = -3.448278556 + 2.625 n = -3.448278556 + 2.625 Combine like terms: -3.448278556 + 2.625 = -0.823278556 n = -0.823278556 Simplifying n = -0.823278556

Solution

The solution to the problem is based on the solutions from the subproblems. n = {6.073278556, -0.823278556}

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