-4n^2+n+10=0

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Solution for -4n^2+n+10=0 equation:


Simplifying
-4n2 + n + 10 = 0

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

Solving
10 + n + -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'.
-2.5 + -0.25n + n2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + -0.25n + 2.5 + n2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + -0.25n + n2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + -0.25n + n2 = 0 + 2.5
-0.25n + n2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
-0.25n + n2 = 2.5

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.25n + 0.25 + n2 = 2.5 + 0.25

Reorder the terms:
0.25 + -0.25n + n2 = 2.5 + 0.25

Combine like terms: 2.5 + 0.25 = 2.75
0.25 + -0.25n + n2 = 2.75

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 2.75

Calculate the square root of the right side: 1.658312395

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.658312395 and -1.658312395.

Subproblem 1

n + 0.5 = 1.658312395 Simplifying n + 0.5 = 1.658312395 Reorder the terms: 0.5 + n = 1.658312395 Solving 0.5 + n = 1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.658312395 + -0.5 n = 1.658312395 + -0.5 Combine like terms: 1.658312395 + -0.5 = 1.158312395 n = 1.158312395 Simplifying n = 1.158312395

Subproblem 2

n + 0.5 = -1.658312395 Simplifying n + 0.5 = -1.658312395 Reorder the terms: 0.5 + n = -1.658312395 Solving 0.5 + n = -1.658312395 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.658312395 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.658312395 + -0.5 n = -1.658312395 + -0.5 Combine like terms: -1.658312395 + -0.5 = -2.158312395 n = -2.158312395 Simplifying n = -2.158312395

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.158312395, -2.158312395}

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