p^2+3p+1=0

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Solution for p^2+3p+1=0 equation:


Simplifying
p2 + 3p + 1 = 0

Reorder the terms:
1 + 3p + p2 = 0

Solving
1 + 3p + p2 = 0

Solving for variable 'p'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3p + -1 + p2 = 0 + -1

Reorder the terms:
1 + -1 + 3p + p2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3p + p2 = 0 + -1
3p + p2 = 0 + -1

Combine like terms: 0 + -1 = -1
3p + p2 = -1

The p term is 3p.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3p + 2.25 + p2 = -1 + 2.25

Reorder the terms:
2.25 + 3p + p2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3p + p2 = 1.25

Factor a perfect square on the left side:
(p + 1.5)(p + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(p + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1

p + 1.5 = 1.118033989 Simplifying p + 1.5 = 1.118033989 Reorder the terms: 1.5 + p = 1.118033989 Solving 1.5 + p = 1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = 1.118033989 + -1.5 p = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 p = -0.381966011 Simplifying p = -0.381966011

Subproblem 2

p + 1.5 = -1.118033989 Simplifying p + 1.5 = -1.118033989 Reorder the terms: 1.5 + p = -1.118033989 Solving 1.5 + p = -1.118033989 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + p = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + p = -1.118033989 + -1.5 p = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 p = -2.618033989 Simplifying p = -2.618033989

Solution

The solution to the problem is based on the solutions from the subproblems. p = {-0.381966011, -2.618033989}

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