n^2-22n+102=0

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Solution for n^2-22n+102=0 equation:


Simplifying
n2 + -22n + 102 = 0

Reorder the terms:
102 + -22n + n2 = 0

Solving
102 + -22n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-102' to each side of the equation.
102 + -22n + -102 + n2 = 0 + -102

Reorder the terms:
102 + -102 + -22n + n2 = 0 + -102

Combine like terms: 102 + -102 = 0
0 + -22n + n2 = 0 + -102
-22n + n2 = 0 + -102

Combine like terms: 0 + -102 = -102
-22n + n2 = -102

The n term is -22n.  Take half its coefficient (-11).
Square it (121) and add it to both sides.

Add '121' to each side of the equation.
-22n + 121 + n2 = -102 + 121

Reorder the terms:
121 + -22n + n2 = -102 + 121

Combine like terms: -102 + 121 = 19
121 + -22n + n2 = 19

Factor a perfect square on the left side:
(n + -11)(n + -11) = 19

Calculate the square root of the right side: 4.358898944

Break this problem into two subproblems by setting 
(n + -11) equal to 4.358898944 and -4.358898944.

Subproblem 1

n + -11 = 4.358898944 Simplifying n + -11 = 4.358898944 Reorder the terms: -11 + n = 4.358898944 Solving -11 + n = 4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = 4.358898944 + 11 Combine like terms: -11 + 11 = 0 0 + n = 4.358898944 + 11 n = 4.358898944 + 11 Combine like terms: 4.358898944 + 11 = 15.358898944 n = 15.358898944 Simplifying n = 15.358898944

Subproblem 2

n + -11 = -4.358898944 Simplifying n + -11 = -4.358898944 Reorder the terms: -11 + n = -4.358898944 Solving -11 + n = -4.358898944 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + n = -4.358898944 + 11 Combine like terms: -11 + 11 = 0 0 + n = -4.358898944 + 11 n = -4.358898944 + 11 Combine like terms: -4.358898944 + 11 = 6.641101056 n = 6.641101056 Simplifying n = 6.641101056

Solution

The solution to the problem is based on the solutions from the subproblems. n = {15.358898944, 6.641101056}

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