6p^2+27p+10=0

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Solution for 6p^2+27p+10=0 equation:


Simplifying
6p2 + 27p + 10 = 0

Reorder the terms:
10 + 27p + 6p2 = 0

Solving
10 + 27p + 6p2 = 0

Solving for variable 'p'.

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

Divide each side by '6'.
1.666666667 + 4.5p + p2 = 0

Move the constant term to the right:

Add '-1.666666667' to each side of the equation.
1.666666667 + 4.5p + -1.666666667 + p2 = 0 + -1.666666667

Reorder the terms:
1.666666667 + -1.666666667 + 4.5p + p2 = 0 + -1.666666667

Combine like terms: 1.666666667 + -1.666666667 = 0.000000000
0.000000000 + 4.5p + p2 = 0 + -1.666666667
4.5p + p2 = 0 + -1.666666667

Combine like terms: 0 + -1.666666667 = -1.666666667
4.5p + p2 = -1.666666667

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 = -1.666666667 + 5.0625

Reorder the terms:
5.0625 + 4.5p + p2 = -1.666666667 + 5.0625

Combine like terms: -1.666666667 + 5.0625 = 3.395833333
5.0625 + 4.5p + p2 = 3.395833333

Factor a perfect square on the left side:
(p + 2.25)(p + 2.25) = 3.395833333

Calculate the square root of the right side: 1.842778699

Break this problem into two subproblems by setting 
(p + 2.25) equal to 1.842778699 and -1.842778699.

Subproblem 1

p + 2.25 = 1.842778699 Simplifying p + 2.25 = 1.842778699 Reorder the terms: 2.25 + p = 1.842778699 Solving 2.25 + p = 1.842778699 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 = 1.842778699 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + p = 1.842778699 + -2.25 p = 1.842778699 + -2.25 Combine like terms: 1.842778699 + -2.25 = -0.407221301 p = -0.407221301 Simplifying p = -0.407221301

Subproblem 2

p + 2.25 = -1.842778699 Simplifying p + 2.25 = -1.842778699 Reorder the terms: 2.25 + p = -1.842778699 Solving 2.25 + p = -1.842778699 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 = -1.842778699 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + p = -1.842778699 + -2.25 p = -1.842778699 + -2.25 Combine like terms: -1.842778699 + -2.25 = -4.092778699 p = -4.092778699 Simplifying p = -4.092778699

Solution

The solution to the problem is based on the solutions from the subproblems. p = {-0.407221301, -4.092778699}

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