9p^2-42p+19=0

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Solution for 9p^2-42p+19=0 equation: 


Simplifying
9p2 + -42p + 19 = 0

Reorder the terms:
19 + -42p + 9p2 = 0

Solving
19 + -42p + 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'.
2.111111111 + -4.666666667p + p2 = 0

Move the constant term to the right:

Add '-2.111111111' to each side of the equation.
2.111111111 + -4.666666667p + -2.111111111 + p2 = 0 + -2.111111111

Reorder the terms:
2.111111111 + -2.111111111 + -4.666666667p + p2 = 0 + -2.111111111

Combine like terms: 2.111111111 + -2.111111111 = 0.000000000
0.000000000 + -4.666666667p + p2 = 0 + -2.111111111
-4.666666667p + p2 = 0 + -2.111111111

Combine like terms: 0 + -2.111111111 = -2.111111111
-4.666666667p + p2 = -2.111111111

The p term is -4.666666667p.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667p + 5.444444448 + p2 = -2.111111111 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667p + p2 = -2.111111111 + 5.444444448

Combine like terms: -2.111111111 + 5.444444448 = 3.333333337
5.444444448 + -4.666666667p + p2 = 3.333333337

Factor a perfect square on the left side:
(p + -2.333333334)(p + -2.333333334) = 3.333333337

Calculate the square root of the right side: 1.825741859

Break this problem into two subproblems by setting 
(p + -2.333333334) equal to 1.825741859 and -1.825741859.


Subproblem 1
p + -2.333333334 = 1.825741859

Simplifying
p + -2.333333334 = 1.825741859

Reorder the terms:
-2.333333334 + p = 1.825741859

Solving
-2.333333334 + p = 1.825741859

Solving for variable 'p'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + p = 1.825741859 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + p = 1.825741859 + 2.333333334
p = 1.825741859 + 2.333333334

Combine like terms: 1.825741859 + 2.333333334 = 4.159075193
p = 4.159075193

Simplifying
p = 4.159075193


Subproblem 2
p + -2.333333334 = -1.825741859

Simplifying
p + -2.333333334 = -1.825741859

Reorder the terms:
-2.333333334 + p = -1.825741859

Solving
-2.333333334 + p = -1.825741859

Solving for variable 'p'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + p = -1.825741859 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + p = -1.825741859 + 2.333333334
p = -1.825741859 + 2.333333334

Combine like terms: -1.825741859 + 2.333333334 = 0.507591475
p = 0.507591475

Simplifying
p = 0.507591475


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {4.159075193, 0.507591475}

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