w^2=7-w

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Solution for w^2=7-w equation:


Simplifying
w2 = 7 + -1w

Solving
w2 = 7 + -1w

Solving for variable 'w'.

Reorder the terms:
-7 + w + w2 = 7 + -1w + -7 + w

Reorder the terms:
-7 + w + w2 = 7 + -7 + -1w + w

Combine like terms: 7 + -7 = 0
-7 + w + w2 = 0 + -1w + w
-7 + w + w2 = -1w + w

Combine like terms: -1w + w = 0
-7 + w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + w + 7 + w2 = 0 + 7

Reorder the terms:
-7 + 7 + w + w2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + w + w2 = 0 + 7
w + w2 = 0 + 7

Combine like terms: 0 + 7 = 7
w + w2 = 7

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.
w + 0.25 + w2 = 7 + 0.25

Reorder the terms:
0.25 + w + w2 = 7 + 0.25

Combine like terms: 7 + 0.25 = 7.25
0.25 + w + w2 = 7.25

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 7.25

Calculate the square root of the right side: 2.692582404

Break this problem into two subproblems by setting 
(w + 0.5) equal to 2.692582404 and -2.692582404.

Subproblem 1

w + 0.5 = 2.692582404 Simplifying w + 0.5 = 2.692582404 Reorder the terms: 0.5 + w = 2.692582404 Solving 0.5 + w = 2.692582404 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 = 2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.692582404 + -0.5 w = 2.692582404 + -0.5 Combine like terms: 2.692582404 + -0.5 = 2.192582404 w = 2.192582404 Simplifying w = 2.192582404

Subproblem 2

w + 0.5 = -2.692582404 Simplifying w + 0.5 = -2.692582404 Reorder the terms: 0.5 + w = -2.692582404 Solving 0.5 + w = -2.692582404 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 = -2.692582404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.692582404 + -0.5 w = -2.692582404 + -0.5 Combine like terms: -2.692582404 + -0.5 = -3.192582404 w = -3.192582404 Simplifying w = -3.192582404

Solution

The solution to the problem is based on the solutions from the subproblems. w = {2.192582404, -3.192582404}

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