z^2-20z=-9

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Solution for z^2-20z=-9 equation:


Simplifying
z2 + -20z = -9

Reorder the terms:
-20z + z2 = -9

Solving
-20z + z2 = -9

Solving for variable 'z'.

Reorder the terms:
9 + -20z + z2 = -9 + 9

Combine like terms: -9 + 9 = 0
9 + -20z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -20z + -9 + z2 = 0 + -9

Reorder the terms:
9 + -9 + -20z + z2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -20z + z2 = 0 + -9
-20z + z2 = 0 + -9

Combine like terms: 0 + -9 = -9
-20z + z2 = -9

The z term is -20z.  Take half its coefficient (-10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
-20z + 100 + z2 = -9 + 100

Reorder the terms:
100 + -20z + z2 = -9 + 100

Combine like terms: -9 + 100 = 91
100 + -20z + z2 = 91

Factor a perfect square on the left side:
(z + -10)(z + -10) = 91

Calculate the square root of the right side: 9.539392014

Break this problem into two subproblems by setting 
(z + -10) equal to 9.539392014 and -9.539392014.

Subproblem 1

z + -10 = 9.539392014 Simplifying z + -10 = 9.539392014 Reorder the terms: -10 + z = 9.539392014 Solving -10 + z = 9.539392014 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + z = 9.539392014 + 10 Combine like terms: -10 + 10 = 0 0 + z = 9.539392014 + 10 z = 9.539392014 + 10 Combine like terms: 9.539392014 + 10 = 19.539392014 z = 19.539392014 Simplifying z = 19.539392014

Subproblem 2

z + -10 = -9.539392014 Simplifying z + -10 = -9.539392014 Reorder the terms: -10 + z = -9.539392014 Solving -10 + z = -9.539392014 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '10' to each side of the equation. -10 + 10 + z = -9.539392014 + 10 Combine like terms: -10 + 10 = 0 0 + z = -9.539392014 + 10 z = -9.539392014 + 10 Combine like terms: -9.539392014 + 10 = 0.460607986 z = 0.460607986 Simplifying z = 0.460607986

Solution

The solution to the problem is based on the solutions from the subproblems. z = {19.539392014, 0.460607986}

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