10p(5p+45)=25

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Solution for 10p(5p+45)=25 equation: 


Simplifying
10p(5p + 45) = 25

Reorder the terms:
10p(45 + 5p) = 25
(45 * 10p + 5p * 10p) = 25
(450p + 50p2) = 25

Solving
450p + 50p2 = 25

Solving for variable 'p'.

Reorder the terms:
-25 + 450p + 50p2 = 25 + -25

Combine like terms: 25 + -25 = 0
-25 + 450p + 50p2 = 0

Factor out the Greatest Common Factor (GCF), '25'.
25(-1 + 18p + 2p2) = 0

Ignore the factor 25.

Subproblem 1
Set the factor '(-1 + 18p + 2p2)' equal to zero and attempt to solve:

Simplifying
-1 + 18p + 2p2 = 0

Solving
-1 + 18p + 2p2 = 0

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

Divide each side by '2'.
-0.5 + 9p + p2 = 0

Move the constant term to the right:

Add '0.5' to each side of the equation.
-0.5 + 9p + 0.5 + p2 = 0 + 0.5

Reorder the terms:
-0.5 + 0.5 + 9p + p2 = 0 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + 9p + p2 = 0 + 0.5
9p + p2 = 0 + 0.5

Combine like terms: 0 + 0.5 = 0.5
9p + p2 = 0.5

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 = 0.5 + 20.25

Reorder the terms:
20.25 + 9p + p2 = 0.5 + 20.25

Combine like terms: 0.5 + 20.25 = 20.75
20.25 + 9p + p2 = 20.75

Factor a perfect square on the left side:
(p + 4.5)(p + 4.5) = 20.75

Calculate the square root of the right side: 4.55521679

Break this problem into two subproblems by setting 
(p + 4.5) equal to 4.55521679 and -4.55521679.


Subproblem 1
p + 4.5 = 4.55521679

Simplifying
p + 4.5 = 4.55521679

Reorder the terms:
4.5 + p = 4.55521679

Solving
4.5 + p = 4.55521679

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 = 4.55521679 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + p = 4.55521679 + -4.5
p = 4.55521679 + -4.5

Combine like terms: 4.55521679 + -4.5 = 0.05521679
p = 0.05521679

Simplifying
p = 0.05521679


Subproblem 2
p + 4.5 = -4.55521679

Simplifying
p + 4.5 = -4.55521679

Reorder the terms:
4.5 + p = -4.55521679

Solving
4.5 + p = -4.55521679

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 = -4.55521679 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + p = -4.55521679 + -4.5
p = -4.55521679 + -4.5

Combine like terms: -4.55521679 + -4.5 = -9.05521679
p = -9.05521679

Simplifying
p = -9.05521679


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {0.05521679, -9.05521679}
Solution
p = {0.05521679, -9.05521679}

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