3+p+2(p-5)p=5

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Solution for 3+p+2(p-5)p=5 equation:


Simplifying
3 + p + 2(p + -5) * p = 5

Reorder the terms:
3 + p + 2(-5 + p) * p = 5

Reorder the terms for easier multiplication:
3 + p + 2p(-5 + p) = 5
3 + p + (-5 * 2p + p * 2p) = 5
3 + p + (-10p + 2p2) = 5

Combine like terms: p + -10p = -9p
3 + -9p + 2p2 = 5

Solving
3 + -9p + 2p2 = 5

Solving for variable 'p'.

Reorder the terms:
3 + -5 + -9p + 2p2 = 5 + -5

Combine like terms: 3 + -5 = -2
-2 + -9p + 2p2 = 5 + -5

Combine like terms: 5 + -5 = 0
-2 + -9p + 2p2 = 0

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

Divide each side by '2'.
-1 + -4.5p + p2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -4.5p + 1 + p2 = 0 + 1

Reorder the terms:
-1 + 1 + -4.5p + p2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -4.5p + p2 = 0 + 1
-4.5p + p2 = 0 + 1

Combine like terms: 0 + 1 = 1
-4.5p + p2 = 1

The p term is -4.5p.  Take half its coefficient (-2.25).
Square it (5.0625) and add it to both sides.

Add '5.0625' to each side of the equation.
-4.5p + 5.0625 + p2 = 1 + 5.0625

Reorder the terms:
5.0625 + -4.5p + p2 = 1 + 5.0625

Combine like terms: 1 + 5.0625 = 6.0625
5.0625 + -4.5p + p2 = 6.0625

Factor a perfect square on the left side:
(p + -2.25)(p + -2.25) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(p + -2.25) equal to 2.46221445 and -2.46221445.

Subproblem 1

p + -2.25 = 2.46221445 Simplifying p + -2.25 = 2.46221445 Reorder the terms: -2.25 + p = 2.46221445 Solving -2.25 + p = 2.46221445 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + p = 2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + p = 2.46221445 + 2.25 p = 2.46221445 + 2.25 Combine like terms: 2.46221445 + 2.25 = 4.71221445 p = 4.71221445 Simplifying p = 4.71221445

Subproblem 2

p + -2.25 = -2.46221445 Simplifying p + -2.25 = -2.46221445 Reorder the terms: -2.25 + p = -2.46221445 Solving -2.25 + p = -2.46221445 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.25' to each side of the equation. -2.25 + 2.25 + p = -2.46221445 + 2.25 Combine like terms: -2.25 + 2.25 = 0.00 0.00 + p = -2.46221445 + 2.25 p = -2.46221445 + 2.25 Combine like terms: -2.46221445 + 2.25 = -0.21221445 p = -0.21221445 Simplifying p = -0.21221445

Solution

The solution to the problem is based on the solutions from the subproblems. p = {4.71221445, -0.21221445}

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