z^2-18z=31

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Solution for z^2-18z=31 equation:


Simplifying
z2 + -18z = 31

Reorder the terms:
-18z + z2 = 31

Solving
-18z + z2 = 31

Solving for variable 'z'.

Reorder the terms:
-31 + -18z + z2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + -18z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -18z + 31 + z2 = 0 + 31

Reorder the terms:
-31 + 31 + -18z + z2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -18z + z2 = 0 + 31
-18z + z2 = 0 + 31

Combine like terms: 0 + 31 = 31
-18z + z2 = 31

The z term is -18z.  Take half its coefficient (-9).
Square it (81) and add it to both sides.

Add '81' to each side of the equation.
-18z + 81 + z2 = 31 + 81

Reorder the terms:
81 + -18z + z2 = 31 + 81

Combine like terms: 31 + 81 = 112
81 + -18z + z2 = 112

Factor a perfect square on the left side:
(z + -9)(z + -9) = 112

Calculate the square root of the right side: 10.583005244

Break this problem into two subproblems by setting 
(z + -9) equal to 10.583005244 and -10.583005244.

Subproblem 1

z + -9 = 10.583005244 Simplifying z + -9 = 10.583005244 Reorder the terms: -9 + z = 10.583005244 Solving -9 + z = 10.583005244 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + z = 10.583005244 + 9 Combine like terms: -9 + 9 = 0 0 + z = 10.583005244 + 9 z = 10.583005244 + 9 Combine like terms: 10.583005244 + 9 = 19.583005244 z = 19.583005244 Simplifying z = 19.583005244

Subproblem 2

z + -9 = -10.583005244 Simplifying z + -9 = -10.583005244 Reorder the terms: -9 + z = -10.583005244 Solving -9 + z = -10.583005244 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '9' to each side of the equation. -9 + 9 + z = -10.583005244 + 9 Combine like terms: -9 + 9 = 0 0 + z = -10.583005244 + 9 z = -10.583005244 + 9 Combine like terms: -10.583005244 + 9 = -1.583005244 z = -1.583005244 Simplifying z = -1.583005244

Solution

The solution to the problem is based on the solutions from the subproblems. z = {19.583005244, -1.583005244}

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