z^2-10z+18=0

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Solution for z^2-10z+18=0 equation:


Simplifying
z2 + -10z + 18 = 0

Reorder the terms:
18 + -10z + z2 = 0

Solving
18 + -10z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-18' to each side of the equation.
18 + -10z + -18 + z2 = 0 + -18

Reorder the terms:
18 + -18 + -10z + z2 = 0 + -18

Combine like terms: 18 + -18 = 0
0 + -10z + z2 = 0 + -18
-10z + z2 = 0 + -18

Combine like terms: 0 + -18 = -18
-10z + z2 = -18

The z term is -10z.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10z + 25 + z2 = -18 + 25

Reorder the terms:
25 + -10z + z2 = -18 + 25

Combine like terms: -18 + 25 = 7
25 + -10z + z2 = 7

Factor a perfect square on the left side:
(z + -5)(z + -5) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(z + -5) equal to 2.645751311 and -2.645751311.

Subproblem 1

z + -5 = 2.645751311 Simplifying z + -5 = 2.645751311 Reorder the terms: -5 + z = 2.645751311 Solving -5 + z = 2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z = 2.645751311 + 5 Combine like terms: -5 + 5 = 0 0 + z = 2.645751311 + 5 z = 2.645751311 + 5 Combine like terms: 2.645751311 + 5 = 7.645751311 z = 7.645751311 Simplifying z = 7.645751311

Subproblem 2

z + -5 = -2.645751311 Simplifying z + -5 = -2.645751311 Reorder the terms: -5 + z = -2.645751311 Solving -5 + z = -2.645751311 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + z = -2.645751311 + 5 Combine like terms: -5 + 5 = 0 0 + z = -2.645751311 + 5 z = -2.645751311 + 5 Combine like terms: -2.645751311 + 5 = 2.354248689 z = 2.354248689 Simplifying z = 2.354248689

Solution

The solution to the problem is based on the solutions from the subproblems. z = {7.645751311, 2.354248689}

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