-4p^2+4000p+8000=0

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Solution for -4p^2+4000p+8000=0 equation: 


Simplifying
-4p2 + 4000p + 8000 = 0

Reorder the terms:
8000 + 4000p + -4p2 = 0

Solving
8000 + 4000p + -4p2 = 0

Solving for variable 'p'.

Factor out the Greatest Common Factor (GCF), '4'.
4(2000 + 1000p + -1p2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(2000 + 1000p + -1p2)' equal to zero and attempt to solve:

Simplifying
2000 + 1000p + -1p2 = 0

Solving
2000 + 1000p + -1p2 = 0

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

Divide each side by '-1'.
-2000 + -1000p + p2 = 0

Move the constant term to the right:

Add '2000' to each side of the equation.
-2000 + -1000p + 2000 + p2 = 0 + 2000

Reorder the terms:
-2000 + 2000 + -1000p + p2 = 0 + 2000

Combine like terms: -2000 + 2000 = 0
0 + -1000p + p2 = 0 + 2000
-1000p + p2 = 0 + 2000

Combine like terms: 0 + 2000 = 2000
-1000p + p2 = 2000

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

Add '250000' to each side of the equation.
-1000p + 250000 + p2 = 2000 + 250000

Reorder the terms:
250000 + -1000p + p2 = 2000 + 250000

Combine like terms: 2000 + 250000 = 252000
250000 + -1000p + p2 = 252000

Factor a perfect square on the left side:
(p + -500)(p + -500) = 252000

Calculate the square root of the right side: 501.99601592

Break this problem into two subproblems by setting 
(p + -500) equal to 501.99601592 and -501.99601592.


Subproblem 1
p + -500 = 501.99601592

Simplifying
p + -500 = 501.99601592

Reorder the terms:
-500 + p = 501.99601592

Solving
-500 + p = 501.99601592

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '500' to each side of the equation.
-500 + 500 + p = 501.99601592 + 500

Combine like terms: -500 + 500 = 0
0 + p = 501.99601592 + 500
p = 501.99601592 + 500

Combine like terms: 501.99601592 + 500 = 1001.99601592
p = 1001.99601592

Simplifying
p = 1001.99601592


Subproblem 2
p + -500 = -501.99601592

Simplifying
p + -500 = -501.99601592

Reorder the terms:
-500 + p = -501.99601592

Solving
-500 + p = -501.99601592

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '500' to each side of the equation.
-500 + 500 + p = -501.99601592 + 500

Combine like terms: -500 + 500 = 0
0 + p = -501.99601592 + 500
p = -501.99601592 + 500

Combine like terms: -501.99601592 + 500 = -1.99601592
p = -1.99601592

Simplifying
p = -1.99601592


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {1001.99601592, -1.99601592}
Solution
p = {1001.99601592, -1.99601592}

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