p^2-10p=31

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Solution for p^2-10p=31 equation:


Simplifying
p2 + -10p = 31

Reorder the terms:
-10p + p2 = 31

Solving
-10p + p2 = 31

Solving for variable 'p'.

Reorder the terms:
-31 + -10p + p2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + -10p + p2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + -10p + 31 + p2 = 0 + 31

Reorder the terms:
-31 + 31 + -10p + p2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + -10p + p2 = 0 + 31
-10p + p2 = 0 + 31

Combine like terms: 0 + 31 = 31
-10p + p2 = 31

The p term is -10p.  Take half its coefficient (-5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
-10p + 25 + p2 = 31 + 25

Reorder the terms:
25 + -10p + p2 = 31 + 25

Combine like terms: 31 + 25 = 56
25 + -10p + p2 = 56

Factor a perfect square on the left side:
(p + -5)(p + -5) = 56

Calculate the square root of the right side: 7.483314774

Break this problem into two subproblems by setting 
(p + -5) equal to 7.483314774 and -7.483314774.

Subproblem 1

p + -5 = 7.483314774 Simplifying p + -5 = 7.483314774 Reorder the terms: -5 + p = 7.483314774 Solving -5 + p = 7.483314774 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + p = 7.483314774 + 5 Combine like terms: -5 + 5 = 0 0 + p = 7.483314774 + 5 p = 7.483314774 + 5 Combine like terms: 7.483314774 + 5 = 12.483314774 p = 12.483314774 Simplifying p = 12.483314774

Subproblem 2

p + -5 = -7.483314774 Simplifying p + -5 = -7.483314774 Reorder the terms: -5 + p = -7.483314774 Solving -5 + p = -7.483314774 Solving for variable 'p'. Move all terms containing p to the left, all other terms to the right. Add '5' to each side of the equation. -5 + 5 + p = -7.483314774 + 5 Combine like terms: -5 + 5 = 0 0 + p = -7.483314774 + 5 p = -7.483314774 + 5 Combine like terms: -7.483314774 + 5 = -2.483314774 p = -2.483314774 Simplifying p = -2.483314774

Solution

The solution to the problem is based on the solutions from the subproblems. p = {12.483314774, -2.483314774}

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