9p^2-48p+22=0

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Solution for 9p^2-48p+22=0 equation: 


Simplifying
9p2 + -48p + 22 = 0

Reorder the terms:
22 + -48p + 9p2 = 0

Solving
22 + -48p + 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 + -5.333333333p + p2 = 0

Move the constant term to the right:

Add '-2.444444444' to each side of the equation.
2.444444444 + -5.333333333p + -2.444444444 + p2 = 0 + -2.444444444

Reorder the terms:
2.444444444 + -2.444444444 + -5.333333333p + p2 = 0 + -2.444444444

Combine like terms: 2.444444444 + -2.444444444 = 0.000000000
0.000000000 + -5.333333333p + p2 = 0 + -2.444444444
-5.333333333p + p2 = 0 + -2.444444444

Combine like terms: 0 + -2.444444444 = -2.444444444
-5.333333333p + p2 = -2.444444444

The p term is -5.333333333p.  Take half its coefficient (-2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
-5.333333333p + 7.111111113 + p2 = -2.444444444 + 7.111111113

Reorder the terms:
7.111111113 + -5.333333333p + p2 = -2.444444444 + 7.111111113

Combine like terms: -2.444444444 + 7.111111113 = 4.666666669
7.111111113 + -5.333333333p + p2 = 4.666666669

Factor a perfect square on the left side:
(p + -2.666666667)(p + -2.666666667) = 4.666666669

Calculate the square root of the right side: 2.1602469

Break this problem into two subproblems by setting 
(p + -2.666666667) equal to 2.1602469 and -2.1602469.


Subproblem 1
p + -2.666666667 = 2.1602469

Simplifying
p + -2.666666667 = 2.1602469

Reorder the terms:
-2.666666667 + p = 2.1602469

Solving
-2.666666667 + p = 2.1602469

Solving for variable 'p'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + p = 2.1602469 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + p = 2.1602469 + 2.666666667
p = 2.1602469 + 2.666666667

Combine like terms: 2.1602469 + 2.666666667 = 4.826913567
p = 4.826913567

Simplifying
p = 4.826913567


Subproblem 2
p + -2.666666667 = -2.1602469

Simplifying
p + -2.666666667 = -2.1602469

Reorder the terms:
-2.666666667 + p = -2.1602469

Solving
-2.666666667 + p = -2.1602469

Solving for variable 'p'.

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

Add '2.666666667' to each side of the equation.
-2.666666667 + 2.666666667 + p = -2.1602469 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + p = -2.1602469 + 2.666666667
p = -2.1602469 + 2.666666667

Combine like terms: -2.1602469 + 2.666666667 = 0.506419767
p = 0.506419767

Simplifying
p = 0.506419767


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {4.826913567, 0.506419767}

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