3.5=w^2+w

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Solution for 3.5=w^2+w equation:


Simplifying
3.5 = w2 + w

Reorder the terms:
3.5 = w + w2

Solving
3.5 = w + w2

Solving for variable 'w'.

Reorder the terms:
3.5 + -1w + -1w2 = w + -1w + w2 + -1w2

Combine like terms: w + -1w = 0
3.5 + -1w + -1w2 = 0 + w2 + -1w2
3.5 + -1w + -1w2 = w2 + -1w2

Combine like terms: w2 + -1w2 = 0
3.5 + -1w + -1w2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3.5 + w + w2 = 0

Move the constant term to the right:

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

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

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + w + w2 = 0 + 3.5
w + w2 = 0 + 3.5

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

The w term is w.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + w2 = 3.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + w2 = 3.5 + 0.25

Combine like terms: 3.5 + 0.25 = 3.75
1.25 + w2 = 3.75

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 3.75

Calculate the square root of the right side: 1.936491673

Break this problem into two subproblems by setting 
(w + 0.5) equal to 1.936491673 and -1.936491673.

Subproblem 1

w + 0.5 = 1.936491673 Simplifying w + 0.5 = 1.936491673 Reorder the terms: 0.5 + w = 1.936491673 Solving 0.5 + w = 1.936491673 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 1.936491673 + -0.5 w = 1.936491673 + -0.5 Combine like terms: 1.936491673 + -0.5 = 1.436491673 w = 1.436491673 Simplifying w = 1.436491673

Subproblem 2

w + 0.5 = -1.936491673 Simplifying w + 0.5 = -1.936491673 Reorder the terms: 0.5 + w = -1.936491673 Solving 0.5 + w = -1.936491673 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -1.936491673 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -1.936491673 + -0.5 w = -1.936491673 + -0.5 Combine like terms: -1.936491673 + -0.5 = -2.436491673 w = -2.436491673 Simplifying w = -2.436491673

Solution

The solution to the problem is based on the solutions from the subproblems. w = {1.436491673, -2.436491673}

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