(p+27)+p(4p-37)=644

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Solution for (p+27)+p(4p-37)=644 equation:


Simplifying
(p + 27) + p(4p + -37) = 644

Reorder the terms:
(27 + p) + p(4p + -37) = 644

Remove parenthesis around (27 + p)
27 + p + p(4p + -37) = 644

Reorder the terms:
27 + p + p(-37 + 4p) = 644
27 + p + (-37 * p + 4p * p) = 644
27 + p + (-37p + 4p2) = 644

Combine like terms: p + -37p = -36p
27 + -36p + 4p2 = 644

Solving
27 + -36p + 4p2 = 644

Solving for variable 'p'.

Reorder the terms:
27 + -644 + -36p + 4p2 = 644 + -644

Combine like terms: 27 + -644 = -617
-617 + -36p + 4p2 = 644 + -644

Combine like terms: 644 + -644 = 0
-617 + -36p + 4p2 = 0

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

Divide each side by '4'.
-154.25 + -9p + p2 = 0

Move the constant term to the right:

Add '154.25' to each side of the equation.
-154.25 + -9p + 154.25 + p2 = 0 + 154.25

Reorder the terms:
-154.25 + 154.25 + -9p + p2 = 0 + 154.25

Combine like terms: -154.25 + 154.25 = 0.00
0.00 + -9p + p2 = 0 + 154.25
-9p + p2 = 0 + 154.25

Combine like terms: 0 + 154.25 = 154.25
-9p + p2 = 154.25

The p term is -9p.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9p + 20.25 + p2 = 154.25 + 20.25

Reorder the terms:
20.25 + -9p + p2 = 154.25 + 20.25

Combine like terms: 154.25 + 20.25 = 174.5
20.25 + -9p + p2 = 174.5

Factor a perfect square on the left side:
(p + -4.5)(p + -4.5) = 174.5

Calculate the square root of the right side: 13.209844814

Break this problem into two subproblems by setting 
(p + -4.5) equal to 13.209844814 and -13.209844814.

Subproblem 1

p + -4.5 = 13.209844814 Simplifying p + -4.5 = 13.209844814 Reorder the terms: -4.5 + p = 13.209844814 Solving -4.5 + p = 13.209844814 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + p = 13.209844814 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + p = 13.209844814 + 4.5 p = 13.209844814 + 4.5 Combine like terms: 13.209844814 + 4.5 = 17.709844814 p = 17.709844814 Simplifying p = 17.709844814

Subproblem 2

p + -4.5 = -13.209844814 Simplifying p + -4.5 = -13.209844814 Reorder the terms: -4.5 + p = -13.209844814 Solving -4.5 + p = -13.209844814 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + p = -13.209844814 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + p = -13.209844814 + 4.5 p = -13.209844814 + 4.5 Combine like terms: -13.209844814 + 4.5 = -8.709844814 p = -8.709844814 Simplifying p = -8.709844814

Solution

The solution to the problem is based on the solutions from the subproblems. p = {17.709844814, -8.709844814}

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