2p^2+pd+9d^2=0

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Solution for 2p^2+pd+9d^2=0 equation: 


Simplifying
2p2 + pd + 9d2 = 0

Reorder the terms:
dp + 9d2 + 2p2 = 0

Solving
dp + 9d2 + 2p2 = 0

Solving for variable 'd'.

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

Divide each side by '9'.
0.1111111111dp + d2 + 0.2222222222p2 = 0

Move the constant term to the right:

Add '-0.2222222222p2' to each side of the equation.
0.1111111111dp + d2 + 0.2222222222p2 + -0.2222222222p2 = 0 + -0.2222222222p2

Combine like terms: 0.2222222222p2 + -0.2222222222p2 = 0.0000000000
0.1111111111dp + d2 + 0.0000000000 = 0 + -0.2222222222p2
0.1111111111dp + d2 = 0 + -0.2222222222p2
Remove the zero:
0.1111111111dp + d2 = -0.2222222222p2

The d term is dp.  Take half its coefficient (0.5p).
Square it (0.25p2) and add it to both sides.

Add '0.25p2' to each side of the equation.
0.1111111111dp + d2 + 0.25p2 = -0.2222222222p2 + 0.25p2

Combine like terms: -0.2222222222p2 + 0.25p2 = 0.0277777778p2
0.1111111111dp + d2 + 0.25p2 = 0.0277777778p2

Factor a perfect square on the left side:
(d + 0.5p)(d + 0.5p) = 0.0277777778p2

Calculate the square root of the right side: 0.166666667p

Break this problem into two subproblems by setting 
(d + 0.5p) equal to 0.166666667p and -0.166666667p.


Subproblem 1
d + 0.5p = 0.166666667p

Simplifying
d + 0.5p = 0.166666667p

Solving
d + 0.5p = 0.166666667p

Solving for variable 'd'.

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

Add '-0.5p' to each side of the equation.
d + 0.5p + -0.5p = 0.166666667p + -0.5p

Combine like terms: 0.5p + -0.5p = 0.0
d + 0.0 = 0.166666667p + -0.5p
d = 0.166666667p + -0.5p

Combine like terms: 0.166666667p + -0.5p = -0.333333333p
d = -0.333333333p

Simplifying
d = -0.333333333p


Subproblem 2
d + 0.5p = -0.166666667p

Simplifying
d + 0.5p = -0.166666667p

Solving
d + 0.5p = -0.166666667p

Solving for variable 'd'.

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

Add '-0.5p' to each side of the equation.
d + 0.5p + -0.5p = -0.166666667p + -0.5p

Combine like terms: 0.5p + -0.5p = 0.0
d + 0.0 = -0.166666667p + -0.5p
d = -0.166666667p + -0.5p

Combine like terms: -0.166666667p + -0.5p = -0.666666667p
d = -0.666666667p

Simplifying
d = -0.666666667p


Solution
The solution to the problem is based on the solutions
from the subproblems.
d = {-0.333333333p, -0.666666667p}

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