4n^2+5n-420=0

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Solution for 4n^2+5n-420=0 equation:


Simplifying
4n2 + 5n + -420 = 0

Reorder the terms:
-420 + 5n + 4n2 = 0

Solving
-420 + 5n + 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'.
-105 + 1.25n + n2 = 0

Move the constant term to the right:

Add '105' to each side of the equation.
-105 + 1.25n + 105 + n2 = 0 + 105

Reorder the terms:
-105 + 105 + 1.25n + n2 = 0 + 105

Combine like terms: -105 + 105 = 0
0 + 1.25n + n2 = 0 + 105
1.25n + n2 = 0 + 105

Combine like terms: 0 + 105 = 105
1.25n + n2 = 105

The n term is 1.25n.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25n + 0.390625 + n2 = 105 + 0.390625

Reorder the terms:
0.390625 + 1.25n + n2 = 105 + 0.390625

Combine like terms: 105 + 0.390625 = 105.390625
0.390625 + 1.25n + n2 = 105.390625

Factor a perfect square on the left side:
(n + 0.625)(n + 0.625) = 105.390625

Calculate the square root of the right side: 10.26599362

Break this problem into two subproblems by setting 
(n + 0.625) equal to 10.26599362 and -10.26599362.

Subproblem 1

n + 0.625 = 10.26599362 Simplifying n + 0.625 = 10.26599362 Reorder the terms: 0.625 + n = 10.26599362 Solving 0.625 + n = 10.26599362 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = 10.26599362 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = 10.26599362 + -0.625 n = 10.26599362 + -0.625 Combine like terms: 10.26599362 + -0.625 = 9.64099362 n = 9.64099362 Simplifying n = 9.64099362

Subproblem 2

n + 0.625 = -10.26599362 Simplifying n + 0.625 = -10.26599362 Reorder the terms: 0.625 + n = -10.26599362 Solving 0.625 + n = -10.26599362 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + n = -10.26599362 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + n = -10.26599362 + -0.625 n = -10.26599362 + -0.625 Combine like terms: -10.26599362 + -0.625 = -10.89099362 n = -10.89099362 Simplifying n = -10.89099362

Solution

The solution to the problem is based on the solutions from the subproblems. n = {9.64099362, -10.89099362}

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