w^2+w=5

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Solution for w^2+w=5 equation:


Simplifying
w2 + w = 5

Reorder the terms:
w + w2 = 5

Solving
w + w2 = 5

Solving for variable 'w'.

Reorder the terms:
-5 + w + w2 = 5 + -5

Combine like terms: 5 + -5 = 0
-5 + w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + w + 5 + w2 = 0 + 5

Reorder the terms:
-5 + 5 + w + w2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + w + w2 = 0 + 5
w + w2 = 0 + 5

Combine like terms: 0 + 5 = 5
w + w2 = 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.
w + 0.25 + w2 = 5 + 0.25

Reorder the terms:
0.25 + w + w2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + w + w2 = 5.25

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(w + 0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1

w + 0.5 = 2.291287847 Simplifying w + 0.5 = 2.291287847 Reorder the terms: 0.5 + w = 2.291287847 Solving 0.5 + w = 2.291287847 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 = 2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 2.291287847 + -0.5 w = 2.291287847 + -0.5 Combine like terms: 2.291287847 + -0.5 = 1.791287847 w = 1.791287847 Simplifying w = 1.791287847

Subproblem 2

w + 0.5 = -2.291287847 Simplifying w + 0.5 = -2.291287847 Reorder the terms: 0.5 + w = -2.291287847 Solving 0.5 + w = -2.291287847 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 = -2.291287847 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -2.291287847 + -0.5 w = -2.291287847 + -0.5 Combine like terms: -2.291287847 + -0.5 = -2.791287847 w = -2.791287847 Simplifying w = -2.791287847

Solution

The solution to the problem is based on the solutions from the subproblems. w = {1.791287847, -2.791287847}

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