4n^2-9n-4=0

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Solution for 4n^2-9n-4=0 equation:


Simplifying
4n2 + -9n + -4 = 0

Reorder the terms:
-4 + -9n + 4n2 = 0

Solving
-4 + -9n + 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 + -2.25n + n2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -2.25n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + -2.25n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2.25n + n2 = 0 + 1
-2.25n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
-2.25n + n2 = 1

The n term is -2.25n.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25n + 1.265625 + n2 = 1 + 1.265625

Reorder the terms:
1.265625 + -2.25n + n2 = 1 + 1.265625

Combine like terms: 1 + 1.265625 = 2.265625
1.265625 + -2.25n + n2 = 2.265625

Factor a perfect square on the left side:
(n + -1.125)(n + -1.125) = 2.265625

Calculate the square root of the right side: 1.505199322

Break this problem into two subproblems by setting 
(n + -1.125) equal to 1.505199322 and -1.505199322.

Subproblem 1

n + -1.125 = 1.505199322 Simplifying n + -1.125 = 1.505199322 Reorder the terms: -1.125 + n = 1.505199322 Solving -1.125 + n = 1.505199322 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + n = 1.505199322 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + n = 1.505199322 + 1.125 n = 1.505199322 + 1.125 Combine like terms: 1.505199322 + 1.125 = 2.630199322 n = 2.630199322 Simplifying n = 2.630199322

Subproblem 2

n + -1.125 = -1.505199322 Simplifying n + -1.125 = -1.505199322 Reorder the terms: -1.125 + n = -1.505199322 Solving -1.125 + n = -1.505199322 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1.125' to each side of the equation. -1.125 + 1.125 + n = -1.505199322 + 1.125 Combine like terms: -1.125 + 1.125 = 0.000 0.000 + n = -1.505199322 + 1.125 n = -1.505199322 + 1.125 Combine like terms: -1.505199322 + 1.125 = -0.380199322 n = -0.380199322 Simplifying n = -0.380199322

Solution

The solution to the problem is based on the solutions from the subproblems. n = {2.630199322, -0.380199322}

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