n^2+n-78=0

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Solution for n^2+n-78=0 equation:


Simplifying
n2 + n + -78 = 0

Reorder the terms:
-78 + n + n2 = 0

Solving
-78 + n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + n + 78 + n2 = 0 + 78

Reorder the terms:
-78 + 78 + n + n2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + n + n2 = 0 + 78
n + n2 = 0 + 78

Combine like terms: 0 + 78 = 78
n + n2 = 78

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.
n + 0.25 + n2 = 78 + 0.25

Reorder the terms:
0.25 + n + n2 = 78 + 0.25

Combine like terms: 78 + 0.25 = 78.25
0.25 + n + n2 = 78.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 78.25

Calculate the square root of the right side: 8.845903006

Break this problem into two subproblems by setting 
(n + 0.5) equal to 8.845903006 and -8.845903006.

Subproblem 1

n + 0.5 = 8.845903006 Simplifying n + 0.5 = 8.845903006 Reorder the terms: 0.5 + n = 8.845903006 Solving 0.5 + n = 8.845903006 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 = 8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 8.845903006 + -0.5 n = 8.845903006 + -0.5 Combine like terms: 8.845903006 + -0.5 = 8.345903006 n = 8.345903006 Simplifying n = 8.345903006

Subproblem 2

n + 0.5 = -8.845903006 Simplifying n + 0.5 = -8.845903006 Reorder the terms: 0.5 + n = -8.845903006 Solving 0.5 + n = -8.845903006 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 = -8.845903006 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -8.845903006 + -0.5 n = -8.845903006 + -0.5 Combine like terms: -8.845903006 + -0.5 = -9.345903006 n = -9.345903006 Simplifying n = -9.345903006

Solution

The solution to the problem is based on the solutions from the subproblems. n = {8.345903006, -9.345903006}

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