n^2=-9n+14

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Solution for n^2=-9n+14 equation:


Simplifying
n2 = -9n + 14

Reorder the terms:
n2 = 14 + -9n

Solving
n2 = 14 + -9n

Solving for variable 'n'.

Reorder the terms:
-14 + 9n + n2 = 14 + -9n + -14 + 9n

Reorder the terms:
-14 + 9n + n2 = 14 + -14 + -9n + 9n

Combine like terms: 14 + -14 = 0
-14 + 9n + n2 = 0 + -9n + 9n
-14 + 9n + n2 = -9n + 9n

Combine like terms: -9n + 9n = 0
-14 + 9n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 9n + 14 + n2 = 0 + 14

Reorder the terms:
-14 + 14 + 9n + n2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 9n + n2 = 0 + 14
9n + n2 = 0 + 14

Combine like terms: 0 + 14 = 14
9n + n2 = 14

The n term is 9n.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9n + 20.25 + n2 = 14 + 20.25

Reorder the terms:
20.25 + 9n + n2 = 14 + 20.25

Combine like terms: 14 + 20.25 = 34.25
20.25 + 9n + n2 = 34.25

Factor a perfect square on the left side:
(n + 4.5)(n + 4.5) = 34.25

Calculate the square root of the right side: 5.852349955

Break this problem into two subproblems by setting 
(n + 4.5) equal to 5.852349955 and -5.852349955.

Subproblem 1

n + 4.5 = 5.852349955 Simplifying n + 4.5 = 5.852349955 Reorder the terms: 4.5 + n = 5.852349955 Solving 4.5 + n = 5.852349955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = 5.852349955 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = 5.852349955 + -4.5 n = 5.852349955 + -4.5 Combine like terms: 5.852349955 + -4.5 = 1.352349955 n = 1.352349955 Simplifying n = 1.352349955

Subproblem 2

n + 4.5 = -5.852349955 Simplifying n + 4.5 = -5.852349955 Reorder the terms: 4.5 + n = -5.852349955 Solving 4.5 + n = -5.852349955 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + n = -5.852349955 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + n = -5.852349955 + -4.5 n = -5.852349955 + -4.5 Combine like terms: -5.852349955 + -4.5 = -10.352349955 n = -10.352349955 Simplifying n = -10.352349955

Solution

The solution to the problem is based on the solutions from the subproblems. n = {1.352349955, -10.352349955}

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