p(3p+14)=15

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Solution for p(3p+14)=15 equation:


Simplifying
p(3p + 14) = 15

Reorder the terms:
p(14 + 3p) = 15
(14 * p + 3p * p) = 15
(14p + 3p2) = 15

Solving
14p + 3p2 = 15

Solving for variable 'p'.

Reorder the terms:
-15 + 14p + 3p2 = 15 + -15

Combine like terms: 15 + -15 = 0
-15 + 14p + 3p2 = 0

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

Divide each side by '3'.
-5 + 4.666666667p + p2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + 4.666666667p + 5 + p2 = 0 + 5

Reorder the terms:
-5 + 5 + 4.666666667p + p2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + 4.666666667p + p2 = 0 + 5
4.666666667p + p2 = 0 + 5

Combine like terms: 0 + 5 = 5
4.666666667p + p2 = 5

The p term is 4.666666667p.  Take half its coefficient (2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
4.666666667p + 5.444444448 + p2 = 5 + 5.444444448

Reorder the terms:
5.444444448 + 4.666666667p + p2 = 5 + 5.444444448

Combine like terms: 5 + 5.444444448 = 10.444444448
5.444444448 + 4.666666667p + p2 = 10.444444448

Factor a perfect square on the left side:
(p + 2.333333334)(p + 2.333333334) = 10.444444448

Calculate the square root of the right side: 3.231786572

Break this problem into two subproblems by setting 
(p + 2.333333334) equal to 3.231786572 and -3.231786572.

Subproblem 1

p + 2.333333334 = 3.231786572 Simplifying p + 2.333333334 = 3.231786572 Reorder the terms: 2.333333334 + p = 3.231786572 Solving 2.333333334 + p = 3.231786572 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + p = 3.231786572 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + p = 3.231786572 + -2.333333334 p = 3.231786572 + -2.333333334 Combine like terms: 3.231786572 + -2.333333334 = 0.898453238 p = 0.898453238 Simplifying p = 0.898453238

Subproblem 2

p + 2.333333334 = -3.231786572 Simplifying p + 2.333333334 = -3.231786572 Reorder the terms: 2.333333334 + p = -3.231786572 Solving 2.333333334 + p = -3.231786572 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + p = -3.231786572 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + p = -3.231786572 + -2.333333334 p = -3.231786572 + -2.333333334 Combine like terms: -3.231786572 + -2.333333334 = -5.565119906 p = -5.565119906 Simplifying p = -5.565119906

Solution

The solution to the problem is based on the solutions from the subproblems. p = {0.898453238, -5.565119906}

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