7p^2+24p+16=0

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Solution for 7p^2+24p+16=0 equation:


Simplifying
7p2 + 24p + 16 = 0

Reorder the terms:
16 + 24p + 7p2 = 0

Solving
16 + 24p + 7p2 = 0

Solving for variable 'p'.

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

Divide each side by '7'.
2.285714286 + 3.428571429p + p2 = 0

Move the constant term to the right:

Add '-2.285714286' to each side of the equation.
2.285714286 + 3.428571429p + -2.285714286 + p2 = 0 + -2.285714286

Reorder the terms:
2.285714286 + -2.285714286 + 3.428571429p + p2 = 0 + -2.285714286

Combine like terms: 2.285714286 + -2.285714286 = 0.000000000
0.000000000 + 3.428571429p + p2 = 0 + -2.285714286
3.428571429p + p2 = 0 + -2.285714286

Combine like terms: 0 + -2.285714286 = -2.285714286
3.428571429p + p2 = -2.285714286

The p term is 3.428571429p.  Take half its coefficient (1.714285715).
Square it (2.938775513) and add it to both sides.

Add '2.938775513' to each side of the equation.
3.428571429p + 2.938775513 + p2 = -2.285714286 + 2.938775513

Reorder the terms:
2.938775513 + 3.428571429p + p2 = -2.285714286 + 2.938775513

Combine like terms: -2.285714286 + 2.938775513 = 0.653061227
2.938775513 + 3.428571429p + p2 = 0.653061227

Factor a perfect square on the left side:
(p + 1.714285715)(p + 1.714285715) = 0.653061227

Calculate the square root of the right side: 0.808122037

Break this problem into two subproblems by setting 
(p + 1.714285715) equal to 0.808122037 and -0.808122037.

Subproblem 1

p + 1.714285715 = 0.808122037 Simplifying p + 1.714285715 = 0.808122037 Reorder the terms: 1.714285715 + p = 0.808122037 Solving 1.714285715 + p = 0.808122037 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + p = 0.808122037 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + p = 0.808122037 + -1.714285715 p = 0.808122037 + -1.714285715 Combine like terms: 0.808122037 + -1.714285715 = -0.906163678 p = -0.906163678 Simplifying p = -0.906163678

Subproblem 2

p + 1.714285715 = -0.808122037 Simplifying p + 1.714285715 = -0.808122037 Reorder the terms: 1.714285715 + p = -0.808122037 Solving 1.714285715 + p = -0.808122037 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + p = -0.808122037 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + p = -0.808122037 + -1.714285715 p = -0.808122037 + -1.714285715 Combine like terms: -0.808122037 + -1.714285715 = -2.522407752 p = -2.522407752 Simplifying p = -2.522407752

Solution

The solution to the problem is based on the solutions from the subproblems. p = {-0.906163678, -2.522407752}

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