(3.5+w)w=31

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Solution for (3.5+w)w=31 equation:


Simplifying
(3.5 + w) * w = 31

Reorder the terms for easier multiplication:
w(3.5 + w) = 31
(3.5 * w + w * w) = 31
(3.5w + w2) = 31

Solving
3.5w + w2 = 31

Solving for variable 'w'.

Reorder the terms:
-31 + 3.5w + w2 = 31 + -31

Combine like terms: 31 + -31 = 0
-31 + 3.5w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '31' to each side of the equation.
-31 + 3.5w + 31 + w2 = 0 + 31

Reorder the terms:
-31 + 31 + 3.5w + w2 = 0 + 31

Combine like terms: -31 + 31 = 0
0 + 3.5w + w2 = 0 + 31
3.5w + w2 = 0 + 31

Combine like terms: 0 + 31 = 31
3.5w + w2 = 31

The w term is 3.5w.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5w + 3.0625 + w2 = 31 + 3.0625

Reorder the terms:
3.0625 + 3.5w + w2 = 31 + 3.0625

Combine like terms: 31 + 3.0625 = 34.0625
3.0625 + 3.5w + w2 = 34.0625

Factor a perfect square on the left side:
(w + 1.75)(w + 1.75) = 34.0625

Calculate the square root of the right side: 5.836308765

Break this problem into two subproblems by setting 
(w + 1.75) equal to 5.836308765 and -5.836308765.

Subproblem 1

w + 1.75 = 5.836308765 Simplifying w + 1.75 = 5.836308765 Reorder the terms: 1.75 + w = 5.836308765 Solving 1.75 + w = 5.836308765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = 5.836308765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = 5.836308765 + -1.75 w = 5.836308765 + -1.75 Combine like terms: 5.836308765 + -1.75 = 4.086308765 w = 4.086308765 Simplifying w = 4.086308765

Subproblem 2

w + 1.75 = -5.836308765 Simplifying w + 1.75 = -5.836308765 Reorder the terms: 1.75 + w = -5.836308765 Solving 1.75 + w = -5.836308765 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + w = -5.836308765 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + w = -5.836308765 + -1.75 w = -5.836308765 + -1.75 Combine like terms: -5.836308765 + -1.75 = -7.586308765 w = -7.586308765 Simplifying w = -7.586308765

Solution

The solution to the problem is based on the solutions from the subproblems. w = {4.086308765, -7.586308765}

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