4p^2-20p+19=0

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Solution for 4p^2-20p+19=0 equation:


Simplifying
4p2 + -20p + 19 = 0

Reorder the terms:
19 + -20p + 4p2 = 0

Solving
19 + -20p + 4p2 = 0

Solving for variable 'p'.

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

Divide each side by '4'.
4.75 + -5p + p2 = 0

Move the constant term to the right:

Add '-4.75' to each side of the equation.
4.75 + -5p + -4.75 + p2 = 0 + -4.75

Reorder the terms:
4.75 + -4.75 + -5p + p2 = 0 + -4.75

Combine like terms: 4.75 + -4.75 = 0.00
0.00 + -5p + p2 = 0 + -4.75
-5p + p2 = 0 + -4.75

Combine like terms: 0 + -4.75 = -4.75
-5p + p2 = -4.75

The p term is -5p.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5p + 6.25 + p2 = -4.75 + 6.25

Reorder the terms:
6.25 + -5p + p2 = -4.75 + 6.25

Combine like terms: -4.75 + 6.25 = 1.5
6.25 + -5p + p2 = 1.5

Factor a perfect square on the left side:
(p + -2.5)(p + -2.5) = 1.5

Calculate the square root of the right side: 1.224744871

Break this problem into two subproblems by setting 
(p + -2.5) equal to 1.224744871 and -1.224744871.

Subproblem 1

p + -2.5 = 1.224744871 Simplifying p + -2.5 = 1.224744871 Reorder the terms: -2.5 + p = 1.224744871 Solving -2.5 + p = 1.224744871 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = 1.224744871 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = 1.224744871 + 2.5 p = 1.224744871 + 2.5 Combine like terms: 1.224744871 + 2.5 = 3.724744871 p = 3.724744871 Simplifying p = 3.724744871

Subproblem 2

p + -2.5 = -1.224744871 Simplifying p + -2.5 = -1.224744871 Reorder the terms: -2.5 + p = -1.224744871 Solving -2.5 + p = -1.224744871 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + p = -1.224744871 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + p = -1.224744871 + 2.5 p = -1.224744871 + 2.5 Combine like terms: -1.224744871 + 2.5 = 1.275255129 p = 1.275255129 Simplifying p = 1.275255129

Solution

The solution to the problem is based on the solutions from the subproblems. p = {3.724744871, 1.275255129}

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