w^2+12w=31

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Solution for w^2+12w=31 equation:


Simplifying
w2 + 12w = 31

Reorder the terms:
12w + w2 = 31

Solving
12w + w2 = 31

Solving for variable 'w'.

Reorder the terms:
-31 + 12w + w2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + 12w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 12w + 31 + w2 = 0 + 31

Reorder the terms:
-31 + 31 + 12w + w2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 12w + w2 = 0 + 31
12w + w2 = 0 + 31

Combine like terms: 0 + 31 = 31
12w + w2 = 31

The w term is 12w.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12w + 36 + w2 = 31 + 36

Reorder the terms:
36 + 12w + w2 = 31 + 36

Combine like terms: 31 + 36 = 67
36 + 12w + w2 = 67

Factor a perfect square on the left side:
(w + 6)(w + 6) = 67

Calculate the square root of the right side: 8.185352772

Break this problem into two subproblems by setting 
(w + 6) equal to 8.185352772 and -8.185352772.

Subproblem 1

w + 6 = 8.185352772 Simplifying w + 6 = 8.185352772 Reorder the terms: 6 + w = 8.185352772 Solving 6 + w = 8.185352772 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + w = 8.185352772 + -6 Combine like terms: 6 + -6 = 0 0 + w = 8.185352772 + -6 w = 8.185352772 + -6 Combine like terms: 8.185352772 + -6 = 2.185352772 w = 2.185352772 Simplifying w = 2.185352772

Subproblem 2

w + 6 = -8.185352772 Simplifying w + 6 = -8.185352772 Reorder the terms: 6 + w = -8.185352772 Solving 6 + w = -8.185352772 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + w = -8.185352772 + -6 Combine like terms: 6 + -6 = 0 0 + w = -8.185352772 + -6 w = -8.185352772 + -6 Combine like terms: -8.185352772 + -6 = -14.185352772 w = -14.185352772 Simplifying w = -14.185352772

Solution

The solution to the problem is based on the solutions from the subproblems. w = {2.185352772, -14.185352772}

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