2x^2+px-2p^2=0

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Solution for 2x^2+px-2p^2=0 equation: 


Simplifying
2x2 + px + -2p2 = 0

Reorder the terms:
px + -2p2 + 2x2 = 0

Solving
px + -2p2 + 2x2 = 0

Solving for variable 'p'.

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-0.5px + p2 + -1x2 = 0

Move the constant term to the right:

Add 'x2' to each side of the equation.
-0.5px + p2 + -1x2 + x2 = 0 + x2

Combine like terms: -1x2 + x2 = 0
-0.5px + p2 + 0 = 0 + x2
-0.5px + p2 = 0 + x2
Remove the zero:
-0.5px + p2 = x2

The p term is px.  Take half its coefficient (0.5x).
Square it (0.25x2) and add it to both sides.

Add '0.25x2' to each side of the equation.
-0.5px + p2 + 0.25x2 = x2 + 0.25x2

Combine like terms: x2 + 0.25x2 = 1.25x2
-0.5px + p2 + 0.25x2 = 1.25x2

Factor a perfect square on the left side:
(p + 0.5x)(p + 0.5x) = 1.25x2

Calculate the square root of the right side: 1.118033989x

Break this problem into two subproblems by setting 
(p + 0.5x) equal to 1.118033989x and -1.118033989x.


Subproblem 1
p + 0.5x = 1.118033989x

Simplifying
p + 0.5x = 1.118033989x

Solving
p + 0.5x = 1.118033989x

Solving for variable 'p'.

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

Add '-0.5x' to each side of the equation.
p + 0.5x + -0.5x = 1.118033989x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
p + 0.0 = 1.118033989x + -0.5x
p = 1.118033989x + -0.5x

Combine like terms: 1.118033989x + -0.5x = 0.618033989x
p = 0.618033989x

Simplifying
p = 0.618033989x


Subproblem 2
p + 0.5x = -1.118033989x

Simplifying
p + 0.5x = -1.118033989x

Solving
p + 0.5x = -1.118033989x

Solving for variable 'p'.

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

Add '-0.5x' to each side of the equation.
p + 0.5x + -0.5x = -1.118033989x + -0.5x

Combine like terms: 0.5x + -0.5x = 0.0
p + 0.0 = -1.118033989x + -0.5x
p = -1.118033989x + -0.5x

Combine like terms: -1.118033989x + -0.5x = -1.618033989x
p = -1.618033989x

Simplifying
p = -1.618033989x


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {0.618033989x, -1.618033989x}

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