n^2-2n=23

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Solution for n^2-2n=23 equation:


Simplifying
n2 + -2n = 23

Reorder the terms:
-2n + n2 = 23

Solving
-2n + n2 = 23

Solving for variable 'n'.

Reorder the terms:
-23 + -2n + n2 = 23 + -23

Combine like terms: 23 + -23 = 0
-23 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + -2n + 23 + n2 = 0 + 23

Reorder the terms:
-23 + 23 + -2n + n2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + -2n + n2 = 0 + 23
-2n + n2 = 0 + 23

Combine like terms: 0 + 23 = 23
-2n + n2 = 23

The n term is -2n.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2n + 1 + n2 = 23 + 1

Reorder the terms:
1 + -2n + n2 = 23 + 1

Combine like terms: 23 + 1 = 24
1 + -2n + n2 = 24

Factor a perfect square on the left side:
(n + -1)(n + -1) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(n + -1) equal to 4.898979486 and -4.898979486.

Subproblem 1

n + -1 = 4.898979486 Simplifying n + -1 = 4.898979486 Reorder the terms: -1 + n = 4.898979486 Solving -1 + n = 4.898979486 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 4.898979486 + 1 Combine like terms: -1 + 1 = 0 0 + n = 4.898979486 + 1 n = 4.898979486 + 1 Combine like terms: 4.898979486 + 1 = 5.898979486 n = 5.898979486 Simplifying n = 5.898979486

Subproblem 2

n + -1 = -4.898979486 Simplifying n + -1 = -4.898979486 Reorder the terms: -1 + n = -4.898979486 Solving -1 + n = -4.898979486 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -4.898979486 + 1 Combine like terms: -1 + 1 = 0 0 + n = -4.898979486 + 1 n = -4.898979486 + 1 Combine like terms: -4.898979486 + 1 = -3.898979486 n = -3.898979486 Simplifying n = -3.898979486

Solution

The solution to the problem is based on the solutions from the subproblems. n = {5.898979486, -3.898979486}

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