16p(p-2)=7p+4

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Solution for 16p(p-2)=7p+4 equation:


Simplifying
16p(p + -2) = 7p + 4

Reorder the terms:
16p(-2 + p) = 7p + 4
(-2 * 16p + p * 16p) = 7p + 4
(-32p + 16p2) = 7p + 4

Reorder the terms:
-32p + 16p2 = 4 + 7p

Solving
-32p + 16p2 = 4 + 7p

Solving for variable 'p'.

Reorder the terms:
-4 + -32p + -7p + 16p2 = 4 + 7p + -4 + -7p

Combine like terms: -32p + -7p = -39p
-4 + -39p + 16p2 = 4 + 7p + -4 + -7p

Reorder the terms:
-4 + -39p + 16p2 = 4 + -4 + 7p + -7p

Combine like terms: 4 + -4 = 0
-4 + -39p + 16p2 = 0 + 7p + -7p
-4 + -39p + 16p2 = 7p + -7p

Combine like terms: 7p + -7p = 0
-4 + -39p + 16p2 = 0

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

Divide each side by '16'.
-0.25 + -2.4375p + p2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -2.4375p + 0.25 + p2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -2.4375p + p2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -2.4375p + p2 = 0 + 0.25
-2.4375p + p2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-2.4375p + p2 = 0.25

The p term is -2.4375p.  Take half its coefficient (-1.21875).
Square it (1.485351563) and add it to both sides.

Add '1.485351563' to each side of the equation.
-2.4375p + 1.485351563 + p2 = 0.25 + 1.485351563

Reorder the terms:
1.485351563 + -2.4375p + p2 = 0.25 + 1.485351563

Combine like terms: 0.25 + 1.485351563 = 1.735351563
1.485351563 + -2.4375p + p2 = 1.735351563

Factor a perfect square on the left side:
(p + -1.21875)(p + -1.21875) = 1.735351563

Calculate the square root of the right side: 1.317327432

Break this problem into two subproblems by setting 
(p + -1.21875) equal to 1.317327432 and -1.317327432.

Subproblem 1

p + -1.21875 = 1.317327432 Simplifying p + -1.21875 = 1.317327432 Reorder the terms: -1.21875 + p = 1.317327432 Solving -1.21875 + p = 1.317327432 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + p = 1.317327432 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + p = 1.317327432 + 1.21875 p = 1.317327432 + 1.21875 Combine like terms: 1.317327432 + 1.21875 = 2.536077432 p = 2.536077432 Simplifying p = 2.536077432

Subproblem 2

p + -1.21875 = -1.317327432 Simplifying p + -1.21875 = -1.317327432 Reorder the terms: -1.21875 + p = -1.317327432 Solving -1.21875 + p = -1.317327432 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '1.21875' to each side of the equation. -1.21875 + 1.21875 + p = -1.317327432 + 1.21875 Combine like terms: -1.21875 + 1.21875 = 0.00000 0.00000 + p = -1.317327432 + 1.21875 p = -1.317327432 + 1.21875 Combine like terms: -1.317327432 + 1.21875 = -0.098577432 p = -0.098577432 Simplifying p = -0.098577432

Solution

The solution to the problem is based on the solutions from the subproblems. p = {2.536077432, -0.098577432}

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