p[0.5p+3]=21

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Solution for p[0.5p+3]=21 equation:


Simplifying
p[0.5p + 3] = 21

Reorder the terms:
p[3 + 0.5p] = 21
[3 * p + 0.5p * p] = 21
[3p + 0.5p2] = 21

Solving
3p + 0.5p2 = 21

Solving for variable 'p'.

Reorder the terms:
-21 + 3p + 0.5p2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 3p + 0.5p2 = 0

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

Divide each side by '0.5'.
-42 + 6p + p2 = 0

Move the constant term to the right:

Add '42' to each side of the equation.
-42 + 6p + 42 + p2 = 0 + 42

Reorder the terms:
-42 + 42 + 6p + p2 = 0 + 42

Combine like terms: -42 + 42 = 0
0 + 6p + p2 = 0 + 42
6p + p2 = 0 + 42

Combine like terms: 0 + 42 = 42
6p + p2 = 42

The p term is 6p.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6p + 9 + p2 = 42 + 9

Reorder the terms:
9 + 6p + p2 = 42 + 9

Combine like terms: 42 + 9 = 51
9 + 6p + p2 = 51

Factor a perfect square on the left side:
(p + 3)(p + 3) = 51

Calculate the square root of the right side: 7.141428429

Break this problem into two subproblems by setting 
(p + 3) equal to 7.141428429 and -7.141428429.

Subproblem 1

p + 3 = 7.141428429 Simplifying p + 3 = 7.141428429 Reorder the terms: 3 + p = 7.141428429 Solving 3 + p = 7.141428429 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + p = 7.141428429 + -3 Combine like terms: 3 + -3 = 0 0 + p = 7.141428429 + -3 p = 7.141428429 + -3 Combine like terms: 7.141428429 + -3 = 4.141428429 p = 4.141428429 Simplifying p = 4.141428429

Subproblem 2

p + 3 = -7.141428429 Simplifying p + 3 = -7.141428429 Reorder the terms: 3 + p = -7.141428429 Solving 3 + p = -7.141428429 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + p = -7.141428429 + -3 Combine like terms: 3 + -3 = 0 0 + p = -7.141428429 + -3 p = -7.141428429 + -3 Combine like terms: -7.141428429 + -3 = -10.141428429 p = -10.141428429 Simplifying p = -10.141428429

Solution

The solution to the problem is based on the solutions from the subproblems. p = {4.141428429, -10.141428429}

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