2p^2+3p=12

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Solution for 2p^2+3p=12 equation:


Simplifying
2p2 + 3p = 12

Reorder the terms:
3p + 2p2 = 12

Solving
3p + 2p2 = 12

Solving for variable 'p'.

Reorder the terms:
-12 + 3p + 2p2 = 12 + -12

Combine like terms: 12 + -12 = 0
-12 + 3p + 2p2 = 0

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

Divide each side by '2'.
-6 + 1.5p + p2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.5p + 6 + p2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.5p + p2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.5p + p2 = 0 + 6
1.5p + p2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.5p + p2 = 6

The p term is 1.5p.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5p + 0.5625 + p2 = 6 + 0.5625

Reorder the terms:
0.5625 + 1.5p + p2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + 1.5p + p2 = 6.5625

Factor a perfect square on the left side:
(p + 0.75)(p + 0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(p + 0.75) equal to 2.561737691 and -2.561737691.

Subproblem 1

p + 0.75 = 2.561737691 Simplifying p + 0.75 = 2.561737691 Reorder the terms: 0.75 + p = 2.561737691 Solving 0.75 + p = 2.561737691 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = 2.561737691 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = 2.561737691 + -0.75 p = 2.561737691 + -0.75 Combine like terms: 2.561737691 + -0.75 = 1.811737691 p = 1.811737691 Simplifying p = 1.811737691

Subproblem 2

p + 0.75 = -2.561737691 Simplifying p + 0.75 = -2.561737691 Reorder the terms: 0.75 + p = -2.561737691 Solving 0.75 + p = -2.561737691 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + p = -2.561737691 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + p = -2.561737691 + -0.75 p = -2.561737691 + -0.75 Combine like terms: -2.561737691 + -0.75 = -3.311737691 p = -3.311737691 Simplifying p = -3.311737691

Solution

The solution to the problem is based on the solutions from the subproblems. p = {1.811737691, -3.311737691}

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