3p^2-9p+1=0

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Solution for 3p^2-9p+1=0 equation:


Simplifying
3p2 + -9p + 1 = 0

Reorder the terms:
1 + -9p + 3p2 = 0

Solving
1 + -9p + 3p2 = 0

Solving for variable 'p'.

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

Divide each side by '3'.
0.3333333333 + -3p + p2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -3p + -0.3333333333 + p2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -3p + p2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -3p + p2 = 0 + -0.3333333333
-3p + p2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-3p + p2 = -0.3333333333

The p term is -3p.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3p + 2.25 + p2 = -0.3333333333 + 2.25

Reorder the terms:
2.25 + -3p + p2 = -0.3333333333 + 2.25

Combine like terms: -0.3333333333 + 2.25 = 1.9166666667
2.25 + -3p + p2 = 1.9166666667

Factor a perfect square on the left side:
(p + -1.5)(p + -1.5) = 1.9166666667

Calculate the square root of the right side: 1.38443731

Break this problem into two subproblems by setting 
(p + -1.5) equal to 1.38443731 and -1.38443731.

Subproblem 1

p + -1.5 = 1.38443731 Simplifying p + -1.5 = 1.38443731 Reorder the terms: -1.5 + p = 1.38443731 Solving -1.5 + p = 1.38443731 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + p = 1.38443731 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + p = 1.38443731 + 1.5 p = 1.38443731 + 1.5 Combine like terms: 1.38443731 + 1.5 = 2.88443731 p = 2.88443731 Simplifying p = 2.88443731

Subproblem 2

p + -1.5 = -1.38443731 Simplifying p + -1.5 = -1.38443731 Reorder the terms: -1.5 + p = -1.38443731 Solving -1.5 + p = -1.38443731 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + p = -1.38443731 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + p = -1.38443731 + 1.5 p = -1.38443731 + 1.5 Combine like terms: -1.38443731 + 1.5 = 0.11556269 p = 0.11556269 Simplifying p = 0.11556269

Solution

The solution to the problem is based on the solutions from the subproblems. p = {2.88443731, 0.11556269}

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