w^2-12w+31=0

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Solution for w^2-12w+31=0 equation:


Simplifying
w2 + -12w + 31 = 0

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

Solving
31 + -12w + w2 = 0

Solving for variable 'w'.

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 = 5
36 + -12w + w2 = 5

Factor a perfect square on the left side:
(w + -6)(w + -6) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(w + -6) equal to 2.236067978 and -2.236067978.

Subproblem 1

w + -6 = 2.236067978 Simplifying w + -6 = 2.236067978 Reorder the terms: -6 + w = 2.236067978 Solving -6 + w = 2.236067978 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 = 2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + w = 2.236067978 + 6 w = 2.236067978 + 6 Combine like terms: 2.236067978 + 6 = 8.236067978 w = 8.236067978 Simplifying w = 8.236067978

Subproblem 2

w + -6 = -2.236067978 Simplifying w + -6 = -2.236067978 Reorder the terms: -6 + w = -2.236067978 Solving -6 + w = -2.236067978 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 = -2.236067978 + 6 Combine like terms: -6 + 6 = 0 0 + w = -2.236067978 + 6 w = -2.236067978 + 6 Combine like terms: -2.236067978 + 6 = 3.763932022 w = 3.763932022 Simplifying w = 3.763932022

Solution

The solution to the problem is based on the solutions from the subproblems. w = {8.236067978, 3.763932022}

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