(3p-4)(p+3)=15

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Solution for (3p-4)(p+3)=15 equation:


Simplifying
(3p + -4)(p + 3) = 15

Reorder the terms:
(-4 + 3p)(p + 3) = 15

Reorder the terms:
(-4 + 3p)(3 + p) = 15

Multiply (-4 + 3p) * (3 + p)
(-4(3 + p) + 3p * (3 + p)) = 15
((3 * -4 + p * -4) + 3p * (3 + p)) = 15
((-12 + -4p) + 3p * (3 + p)) = 15
(-12 + -4p + (3 * 3p + p * 3p)) = 15
(-12 + -4p + (9p + 3p2)) = 15

Combine like terms: -4p + 9p = 5p
(-12 + 5p + 3p2) = 15

Solving
-12 + 5p + 3p2 = 15

Solving for variable 'p'.

Reorder the terms:
-12 + -15 + 5p + 3p2 = 15 + -15

Combine like terms: -12 + -15 = -27
-27 + 5p + 3p2 = 15 + -15

Combine like terms: 15 + -15 = 0
-27 + 5p + 3p2 = 0

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

Divide each side by '3'.
-9 + 1.666666667p + p2 = 0

Move the constant term to the right:

Add '9' to each side of the equation.
-9 + 1.666666667p + 9 + p2 = 0 + 9

Reorder the terms:
-9 + 9 + 1.666666667p + p2 = 0 + 9

Combine like terms: -9 + 9 = 0
0 + 1.666666667p + p2 = 0 + 9
1.666666667p + p2 = 0 + 9

Combine like terms: 0 + 9 = 9
1.666666667p + p2 = 9

The p term is 1.666666667p.  Take half its coefficient (0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
1.666666667p + 0.6944444447 + p2 = 9 + 0.6944444447

Reorder the terms:
0.6944444447 + 1.666666667p + p2 = 9 + 0.6944444447

Combine like terms: 9 + 0.6944444447 = 9.6944444447
0.6944444447 + 1.666666667p + p2 = 9.6944444447

Factor a perfect square on the left side:
(p + 0.8333333335)(p + 0.8333333335) = 9.6944444447

Calculate the square root of the right side: 3.113590282

Break this problem into two subproblems by setting 
(p + 0.8333333335) equal to 3.113590282 and -3.113590282.

Subproblem 1

p + 0.8333333335 = 3.113590282 Simplifying p + 0.8333333335 = 3.113590282 Reorder the terms: 0.8333333335 + p = 3.113590282 Solving 0.8333333335 + p = 3.113590282 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + p = 3.113590282 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + p = 3.113590282 + -0.8333333335 p = 3.113590282 + -0.8333333335 Combine like terms: 3.113590282 + -0.8333333335 = 2.2802569485 p = 2.2802569485 Simplifying p = 2.2802569485

Subproblem 2

p + 0.8333333335 = -3.113590282 Simplifying p + 0.8333333335 = -3.113590282 Reorder the terms: 0.8333333335 + p = -3.113590282 Solving 0.8333333335 + p = -3.113590282 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.8333333335' to each side of the equation. 0.8333333335 + -0.8333333335 + p = -3.113590282 + -0.8333333335 Combine like terms: 0.8333333335 + -0.8333333335 = 0.0000000000 0.0000000000 + p = -3.113590282 + -0.8333333335 p = -3.113590282 + -0.8333333335 Combine like terms: -3.113590282 + -0.8333333335 = -3.9469236155 p = -3.9469236155 Simplifying p = -3.9469236155

Solution

The solution to the problem is based on the solutions from the subproblems. p = {2.2802569485, -3.9469236155}

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