40p+3*1.49*p*p=257

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Solution for 40p+3*1.49*p*p=257 equation:


Simplifying
40p + 3 * 1.49p * p = 257

Multiply 3 * 1.49
40p + 4.47p * p = 257

Multiply p * p
40p + 4.47p2 = 257

Solving
40p + 4.47p2 = 257

Solving for variable 'p'.

Reorder the terms:
-257 + 40p + 4.47p2 = 257 + -257

Combine like terms: 257 + -257 = 0
-257 + 40p + 4.47p2 = 0

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

Divide each side by '4.47'.
-57.49440716 + 8.948545861p + p2 = 0

Move the constant term to the right:

Add '57.49440716' to each side of the equation.
-57.49440716 + 8.948545861p + 57.49440716 + p2 = 0 + 57.49440716

Reorder the terms:
-57.49440716 + 57.49440716 + 8.948545861p + p2 = 0 + 57.49440716

Combine like terms: -57.49440716 + 57.49440716 = 0.00000000
0.00000000 + 8.948545861p + p2 = 0 + 57.49440716
8.948545861p + p2 = 0 + 57.49440716

Combine like terms: 0 + 57.49440716 = 57.49440716
8.948545861p + p2 = 57.49440716

The p term is 8.948545861p.  Take half its coefficient (4.474272931).
Square it (20.01911826) and add it to both sides.

Add '20.01911826' to each side of the equation.
8.948545861p + 20.01911826 + p2 = 57.49440716 + 20.01911826

Reorder the terms:
20.01911826 + 8.948545861p + p2 = 57.49440716 + 20.01911826

Combine like terms: 57.49440716 + 20.01911826 = 77.51352542
20.01911826 + 8.948545861p + p2 = 77.51352542

Factor a perfect square on the left side:
(p + 4.474272931)(p + 4.474272931) = 77.51352542

Calculate the square root of the right side: 8.80417659

Break this problem into two subproblems by setting 
(p + 4.474272931) equal to 8.80417659 and -8.80417659.

Subproblem 1

p + 4.474272931 = 8.80417659 Simplifying p + 4.474272931 = 8.80417659 Reorder the terms: 4.474272931 + p = 8.80417659 Solving 4.474272931 + p = 8.80417659 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-4.474272931' to each side of the equation. 4.474272931 + -4.474272931 + p = 8.80417659 + -4.474272931 Combine like terms: 4.474272931 + -4.474272931 = 0.000000000 0.000000000 + p = 8.80417659 + -4.474272931 p = 8.80417659 + -4.474272931 Combine like terms: 8.80417659 + -4.474272931 = 4.329903659 p = 4.329903659 Simplifying p = 4.329903659

Subproblem 2

p + 4.474272931 = -8.80417659 Simplifying p + 4.474272931 = -8.80417659 Reorder the terms: 4.474272931 + p = -8.80417659 Solving 4.474272931 + p = -8.80417659 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-4.474272931' to each side of the equation. 4.474272931 + -4.474272931 + p = -8.80417659 + -4.474272931 Combine like terms: 4.474272931 + -4.474272931 = 0.000000000 0.000000000 + p = -8.80417659 + -4.474272931 p = -8.80417659 + -4.474272931 Combine like terms: -8.80417659 + -4.474272931 = -13.278449521 p = -13.278449521 Simplifying p = -13.278449521

Solution

The solution to the problem is based on the solutions from the subproblems. p = {4.329903659, -13.278449521}

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