z^2-1000z+10100=0

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Solution for z^2-1000z+10100=0 equation:


Simplifying
z2 + -1000z + 10100 = 0

Reorder the terms:
10100 + -1000z + z2 = 0

Solving
10100 + -1000z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '-10100' to each side of the equation.
10100 + -1000z + -10100 + z2 = 0 + -10100

Reorder the terms:
10100 + -10100 + -1000z + z2 = 0 + -10100

Combine like terms: 10100 + -10100 = 0
0 + -1000z + z2 = 0 + -10100
-1000z + z2 = 0 + -10100

Combine like terms: 0 + -10100 = -10100
-1000z + z2 = -10100

The z term is -1000z.  Take half its coefficient (-500).
Square it (250000) and add it to both sides.

Add '250000' to each side of the equation.
-1000z + 250000 + z2 = -10100 + 250000

Reorder the terms:
250000 + -1000z + z2 = -10100 + 250000

Combine like terms: -10100 + 250000 = 239900
250000 + -1000z + z2 = 239900

Factor a perfect square on the left side:
(z + -500)(z + -500) = 239900

Calculate the square root of the right side: 489.79587585

Break this problem into two subproblems by setting 
(z + -500) equal to 489.79587585 and -489.79587585.

Subproblem 1

z + -500 = 489.79587585 Simplifying z + -500 = 489.79587585 Reorder the terms: -500 + z = 489.79587585 Solving -500 + z = 489.79587585 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + z = 489.79587585 + 500 Combine like terms: -500 + 500 = 0 0 + z = 489.79587585 + 500 z = 489.79587585 + 500 Combine like terms: 489.79587585 + 500 = 989.79587585 z = 989.79587585 Simplifying z = 989.79587585

Subproblem 2

z + -500 = -489.79587585 Simplifying z + -500 = -489.79587585 Reorder the terms: -500 + z = -489.79587585 Solving -500 + z = -489.79587585 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + z = -489.79587585 + 500 Combine like terms: -500 + 500 = 0 0 + z = -489.79587585 + 500 z = -489.79587585 + 500 Combine like terms: -489.79587585 + 500 = 10.20412415 z = 10.20412415 Simplifying z = 10.20412415

Solution

The solution to the problem is based on the solutions from the subproblems. z = {989.79587585, 10.20412415}

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