2.5p(2p+5)=5p+12.5

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Solution for 2.5p(2p+5)=5p+12.5 equation:


Simplifying
2.5p(2p + 5) = 5p + 12.5

Reorder the terms:
2.5p(5 + 2p) = 5p + 12.5
(5 * 2.5p + 2p * 2.5p) = 5p + 12.5
(12.5p + 5p2) = 5p + 12.5

Reorder the terms:
12.5p + 5p2 = 12.5 + 5p

Solving
12.5p + 5p2 = 12.5 + 5p

Solving for variable 'p'.

Reorder the terms:
-12.5 + 12.5p + -5p + 5p2 = 12.5 + 5p + -12.5 + -5p

Combine like terms: 12.5p + -5p = 7.5p
-12.5 + 7.5p + 5p2 = 12.5 + 5p + -12.5 + -5p

Reorder the terms:
-12.5 + 7.5p + 5p2 = 12.5 + -12.5 + 5p + -5p

Combine like terms: 12.5 + -12.5 = 0.0
-12.5 + 7.5p + 5p2 = 0.0 + 5p + -5p
-12.5 + 7.5p + 5p2 = 5p + -5p

Combine like terms: 5p + -5p = 0
-12.5 + 7.5p + 5p2 = 0

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

Divide each side by '5'.
-2.5 + 1.5p + p2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 1.5p + 2.5 + p2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 1.5p + p2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 1.5p + p2 = 0 + 2.5
1.5p + p2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
1.5p + p2 = 2.5

The p term is 1.5p.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5p + 0.5625 + p2 = 2.5 + 0.5625

Reorder the terms:
0.5625 + 1.5p + p2 = 2.5 + 0.5625

Combine like terms: 2.5 + 0.5625 = 3.0625
0.5625 + 1.5p + p2 = 3.0625

Factor a perfect square on the left side:
(p + 0.75)(p + 0.75) = 3.0625

Calculate the square root of the right side: 1.75

Break this problem into two subproblems by setting 
(p + 0.75) equal to 1.75 and -1.75.

Subproblem 1

p + 0.75 = 1.75 Simplifying p + 0.75 = 1.75 Reorder the terms: 0.75 + p = 1.75 Solving 0.75 + p = 1.75 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = 1.75 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = 1.75 + -0.75 p = 1.75 + -0.75 Combine like terms: 1.75 + -0.75 = 1 p = 1 Simplifying p = 1

Subproblem 2

p + 0.75 = -1.75 Simplifying p + 0.75 = -1.75 Reorder the terms: 0.75 + p = -1.75 Solving 0.75 + p = -1.75 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = -1.75 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = -1.75 + -0.75 p = -1.75 + -0.75 Combine like terms: -1.75 + -0.75 = -2.5 p = -2.5 Simplifying p = -2.5

Solution

The solution to the problem is based on the solutions from the subproblems. p = {1, -2.5}

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