z^2-10z=20

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Solution for z^2-10z=20 equation:


Simplifying
z2 + -10z = 20

Reorder the terms:
-10z + z2 = 20

Solving
-10z + z2 = 20

Solving for variable 'z'.

Reorder the terms:
-20 + -10z + z2 = 20 + -20

Combine like terms: 20 + -20 = 0
-20 + -10z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + -10z + 20 + z2 = 0 + 20

Reorder the terms:
-20 + 20 + -10z + z2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + -10z + z2 = 0 + 20
-10z + z2 = 0 + 20

Combine like terms: 0 + 20 = 20
-10z + z2 = 20

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 = 20 + 25

Reorder the terms:
25 + -10z + z2 = 20 + 25

Combine like terms: 20 + 25 = 45
25 + -10z + z2 = 45

Factor a perfect square on the left side:
(z + -5)(z + -5) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(z + -5) equal to 6.708203933 and -6.708203933.

Subproblem 1

z + -5 = 6.708203933 Simplifying z + -5 = 6.708203933 Reorder the terms: -5 + z = 6.708203933 Solving -5 + z = 6.708203933 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 = 6.708203933 + 5 Combine like terms: -5 + 5 = 0 0 + z = 6.708203933 + 5 z = 6.708203933 + 5 Combine like terms: 6.708203933 + 5 = 11.708203933 z = 11.708203933 Simplifying z = 11.708203933

Subproblem 2

z + -5 = -6.708203933 Simplifying z + -5 = -6.708203933 Reorder the terms: -5 + z = -6.708203933 Solving -5 + z = -6.708203933 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 = -6.708203933 + 5 Combine like terms: -5 + 5 = 0 0 + z = -6.708203933 + 5 z = -6.708203933 + 5 Combine like terms: -6.708203933 + 5 = -1.708203933 z = -1.708203933 Simplifying z = -1.708203933

Solution

The solution to the problem is based on the solutions from the subproblems. z = {11.708203933, -1.708203933}

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