1000z-z^2=10

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Solution for 1000z-z^2=10 equation:


Simplifying
1000z + -1z2 = 10

Solving
1000z + -1z2 = 10

Solving for variable 'z'.

Reorder the terms:
-10 + 1000z + -1z2 = 10 + -10

Combine like terms: 10 + -10 = 0
-10 + 1000z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
10 + -1000z + z2 = 0

Move the constant term to the right:

Add '-10' to each side of the equation.
10 + -1000z + -10 + z2 = 0 + -10

Reorder the terms:
10 + -10 + -1000z + z2 = 0 + -10

Combine like terms: 10 + -10 = 0
0 + -1000z + z2 = 0 + -10
-1000z + z2 = 0 + -10

Combine like terms: 0 + -10 = -10
-1000z + z2 = -10

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 = -10 + 250000

Reorder the terms:
250000 + -1000z + z2 = -10 + 250000

Combine like terms: -10 + 250000 = 249990
250000 + -1000z + z2 = 249990

Factor a perfect square on the left side:
(z + -500)(z + -500) = 249990

Calculate the square root of the right side: 499.9899999

Break this problem into two subproblems by setting 
(z + -500) equal to 499.9899999 and -499.9899999.

Subproblem 1

z + -500 = 499.9899999 Simplifying z + -500 = 499.9899999 Reorder the terms: -500 + z = 499.9899999 Solving -500 + z = 499.9899999 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 = 499.9899999 + 500 Combine like terms: -500 + 500 = 0 0 + z = 499.9899999 + 500 z = 499.9899999 + 500 Combine like terms: 499.9899999 + 500 = 999.9899999 z = 999.9899999 Simplifying z = 999.9899999

Subproblem 2

z + -500 = -499.9899999 Simplifying z + -500 = -499.9899999 Reorder the terms: -500 + z = -499.9899999 Solving -500 + z = -499.9899999 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 = -499.9899999 + 500 Combine like terms: -500 + 500 = 0 0 + z = -499.9899999 + 500 z = -499.9899999 + 500 Combine like terms: -499.9899999 + 500 = 0.0100001 z = 0.0100001 Simplifying z = 0.0100001

Solution

The solution to the problem is based on the solutions from the subproblems. z = {999.9899999, 0.0100001}

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