3p^2+11p=7

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Solution for 3p^2+11p=7 equation: 


Simplifying
3p2 + 11p = 7

Reorder the terms:
11p + 3p2 = 7

Solving
11p + 3p2 = 7

Solving for variable 'p'.

Reorder the terms:
-7 + 11p + 3p2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 11p + 3p2 = 0

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

Divide each side by '3'.
-2.333333333 + 3.666666667p + p2 = 0

Move the constant term to the right:

Add '2.333333333' to each side of the equation.
-2.333333333 + 3.666666667p + 2.333333333 + p2 = 0 + 2.333333333

Reorder the terms:
-2.333333333 + 2.333333333 + 3.666666667p + p2 = 0 + 2.333333333

Combine like terms: -2.333333333 + 2.333333333 = 0.000000000
0.000000000 + 3.666666667p + p2 = 0 + 2.333333333
3.666666667p + p2 = 0 + 2.333333333

Combine like terms: 0 + 2.333333333 = 2.333333333
3.666666667p + p2 = 2.333333333

The p term is 3.666666667p.  Take half its coefficient (1.833333334).
Square it (3.361111114) and add it to both sides.

Add '3.361111114' to each side of the equation.
3.666666667p + 3.361111114 + p2 = 2.333333333 + 3.361111114

Reorder the terms:
3.361111114 + 3.666666667p + p2 = 2.333333333 + 3.361111114

Combine like terms: 2.333333333 + 3.361111114 = 5.694444447
3.361111114 + 3.666666667p + p2 = 5.694444447

Factor a perfect square on the left side:
(p + 1.833333334)(p + 1.833333334) = 5.694444447

Calculate the square root of the right side: 2.386303511

Break this problem into two subproblems by setting 
(p + 1.833333334) equal to 2.386303511 and -2.386303511.


Subproblem 1
p + 1.833333334 = 2.386303511

Simplifying
p + 1.833333334 = 2.386303511

Reorder the terms:
1.833333334 + p = 2.386303511

Solving
1.833333334 + p = 2.386303511

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + p = 2.386303511 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + p = 2.386303511 + -1.833333334
p = 2.386303511 + -1.833333334

Combine like terms: 2.386303511 + -1.833333334 = 0.552970177
p = 0.552970177

Simplifying
p = 0.552970177


Subproblem 2
p + 1.833333334 = -2.386303511

Simplifying
p + 1.833333334 = -2.386303511

Reorder the terms:
1.833333334 + p = -2.386303511

Solving
1.833333334 + p = -2.386303511

Solving for variable 'p'.

Move all terms containing p to the left, all other terms to the right.

Add '-1.833333334' to each side of the equation.
1.833333334 + -1.833333334 + p = -2.386303511 + -1.833333334

Combine like terms: 1.833333334 + -1.833333334 = 0.000000000
0.000000000 + p = -2.386303511 + -1.833333334
p = -2.386303511 + -1.833333334

Combine like terms: -2.386303511 + -1.833333334 = -4.219636845
p = -4.219636845

Simplifying
p = -4.219636845


Solution
The solution to the problem is based on the solutions
from the subproblems.
p = {0.552970177, -4.219636845}

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