z^2-22z=24

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Solution for z^2-22z=24 equation:


Simplifying
z2 + -22z = 24

Reorder the terms:
-22z + z2 = 24

Solving
-22z + z2 = 24

Solving for variable 'z'.

Reorder the terms:
-24 + -22z + z2 = 24 + -24

Combine like terms: 24 + -24 = 0
-24 + -22z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '24' to each side of the equation.
-24 + -22z + 24 + z2 = 0 + 24

Reorder the terms:
-24 + 24 + -22z + z2 = 0 + 24

Combine like terms: -24 + 24 = 0
0 + -22z + z2 = 0 + 24
-22z + z2 = 0 + 24

Combine like terms: 0 + 24 = 24
-22z + z2 = 24

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

Add '121' to each side of the equation.
-22z + 121 + z2 = 24 + 121

Reorder the terms:
121 + -22z + z2 = 24 + 121

Combine like terms: 24 + 121 = 145
121 + -22z + z2 = 145

Factor a perfect square on the left side:
(z + -11)(z + -11) = 145

Calculate the square root of the right side: 12.041594579

Break this problem into two subproblems by setting 
(z + -11) equal to 12.041594579 and -12.041594579.

Subproblem 1

z + -11 = 12.041594579 Simplifying z + -11 = 12.041594579 Reorder the terms: -11 + z = 12.041594579 Solving -11 + z = 12.041594579 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + z = 12.041594579 + 11 Combine like terms: -11 + 11 = 0 0 + z = 12.041594579 + 11 z = 12.041594579 + 11 Combine like terms: 12.041594579 + 11 = 23.041594579 z = 23.041594579 Simplifying z = 23.041594579

Subproblem 2

z + -11 = -12.041594579 Simplifying z + -11 = -12.041594579 Reorder the terms: -11 + z = -12.041594579 Solving -11 + z = -12.041594579 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + z = -12.041594579 + 11 Combine like terms: -11 + 11 = 0 0 + z = -12.041594579 + 11 z = -12.041594579 + 11 Combine like terms: -12.041594579 + 11 = -1.041594579 z = -1.041594579 Simplifying z = -1.041594579

Solution

The solution to the problem is based on the solutions from the subproblems. z = {23.041594579, -1.041594579}

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