3p^2=6p-1

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Solution for 3p^2=6p-1 equation:


Simplifying
3p2 = 6p + -1

Reorder the terms:
3p2 = -1 + 6p

Solving
3p2 = -1 + 6p

Solving for variable 'p'.

Reorder the terms:
1 + -6p + 3p2 = -1 + 6p + 1 + -6p

Reorder the terms:
1 + -6p + 3p2 = -1 + 1 + 6p + -6p

Combine like terms: -1 + 1 = 0
1 + -6p + 3p2 = 0 + 6p + -6p
1 + -6p + 3p2 = 6p + -6p

Combine like terms: 6p + -6p = 0
1 + -6p + 3p2 = 0

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

Divide each side by '3'.
0.3333333333 + -2p + p2 = 0

Move the constant term to the right:

Add '-0.3333333333' to each side of the equation.
0.3333333333 + -2p + -0.3333333333 + p2 = 0 + -0.3333333333

Reorder the terms:
0.3333333333 + -0.3333333333 + -2p + p2 = 0 + -0.3333333333

Combine like terms: 0.3333333333 + -0.3333333333 = 0.0000000000
0.0000000000 + -2p + p2 = 0 + -0.3333333333
-2p + p2 = 0 + -0.3333333333

Combine like terms: 0 + -0.3333333333 = -0.3333333333
-2p + p2 = -0.3333333333

The p term is -2p.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2p + 1 + p2 = -0.3333333333 + 1

Reorder the terms:
1 + -2p + p2 = -0.3333333333 + 1

Combine like terms: -0.3333333333 + 1 = 0.6666666667
1 + -2p + p2 = 0.6666666667

Factor a perfect square on the left side:
(p + -1)(p + -1) = 0.6666666667

Calculate the square root of the right side: 0.816496581

Break this problem into two subproblems by setting 
(p + -1) equal to 0.816496581 and -0.816496581.

Subproblem 1

p + -1 = 0.816496581 Simplifying p + -1 = 0.816496581 Reorder the terms: -1 + p = 0.816496581 Solving -1 + p = 0.816496581 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = 0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + p = 0.816496581 + 1 p = 0.816496581 + 1 Combine like terms: 0.816496581 + 1 = 1.816496581 p = 1.816496581 Simplifying p = 1.816496581

Subproblem 2

p + -1 = -0.816496581 Simplifying p + -1 = -0.816496581 Reorder the terms: -1 + p = -0.816496581 Solving -1 + p = -0.816496581 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + p = -0.816496581 + 1 Combine like terms: -1 + 1 = 0 0 + p = -0.816496581 + 1 p = -0.816496581 + 1 Combine like terms: -0.816496581 + 1 = 0.183503419 p = 0.183503419 Simplifying p = 0.183503419

Solution

The solution to the problem is based on the solutions from the subproblems. p = {1.816496581, 0.183503419}

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