9p^2-30p+22=0

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Solution for 9p^2-30p+22=0 equation: 


Simplifying
9p2 + -30p + 22 = 0

Reorder the terms:
22 + -30p + 9p2 = 0

Solving
22 + -30p + 9p2 = 0

Solving for variable 'p'.

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

Divide each side by '9'.
2.444444444 + -3.333333333p + p2 = 0

Move the constant term to the right:

Add '-2.444444444' to each side of the equation.
2.444444444 + -3.333333333p + -2.444444444 + p2 = 0 + -2.444444444

Reorder the terms:
2.444444444 + -2.444444444 + -3.333333333p + p2 = 0 + -2.444444444

Combine like terms: 2.444444444 + -2.444444444 = 0.000000000
0.000000000 + -3.333333333p + p2 = 0 + -2.444444444
-3.333333333p + p2 = 0 + -2.444444444

Combine like terms: 0 + -2.444444444 = -2.444444444
-3.333333333p + p2 = -2.444444444

The p term is -3.333333333p.  Take half its coefficient (-1.666666667).
Square it (2.777777779) and add it to both sides.

Add '2.777777779' to each side of the equation.
-3.333333333p + 2.777777779 + p2 = -2.444444444 + 2.777777779

Reorder the terms:
2.777777779 + -3.333333333p + p2 = -2.444444444 + 2.777777779

Combine like terms: -2.444444444 + 2.777777779 = 0.333333335
2.777777779 + -3.333333333p + p2 = 0.333333335

Factor a perfect square on the left side:
(p + -1.666666667)(p + -1.666666667) = 0.333333335

Calculate the square root of the right side: 0.577350271

Break this problem into two subproblems by setting 
(p + -1.666666667) equal to 0.577350271 and -0.577350271.


Subproblem 1
p + -1.666666667 = 0.577350271

Simplifying
p + -1.666666667 = 0.577350271

Reorder the terms:
-1.666666667 + p = 0.577350271

Solving
-1.666666667 + p = 0.577350271

Solving for variable 'p'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + p = 0.577350271 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + p = 0.577350271 + 1.666666667
p = 0.577350271 + 1.666666667

Combine like terms: 0.577350271 + 1.666666667 = 2.244016938
p = 2.244016938

Simplifying
p = 2.244016938


Subproblem 2
p + -1.666666667 = -0.577350271

Simplifying
p + -1.666666667 = -0.577350271

Reorder the terms:
-1.666666667 + p = -0.577350271

Solving
-1.666666667 + p = -0.577350271

Solving for variable 'p'.

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

Add '1.666666667' to each side of the equation.
-1.666666667 + 1.666666667 + p = -0.577350271 + 1.666666667

Combine like terms: -1.666666667 + 1.666666667 = 0.000000000
0.000000000 + p = -0.577350271 + 1.666666667
p = -0.577350271 + 1.666666667

Combine like terms: -0.577350271 + 1.666666667 = 1.089316396
p = 1.089316396

Simplifying
p = 1.089316396


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {2.244016938, 1.089316396}

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