w^2+13w=40

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Solution for w^2+13w=40 equation:


Simplifying
w2 + 13w = 40

Reorder the terms:
13w + w2 = 40

Solving
13w + w2 = 40

Solving for variable 'w'.

Reorder the terms:
-40 + 13w + w2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 13w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '40' to each side of the equation.
-40 + 13w + 40 + w2 = 0 + 40

Reorder the terms:
-40 + 40 + 13w + w2 = 0 + 40

Combine like terms: -40 + 40 = 0
0 + 13w + w2 = 0 + 40
13w + w2 = 0 + 40

Combine like terms: 0 + 40 = 40
13w + w2 = 40

The w term is 13w.  Take half its coefficient (6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
13w + 42.25 + w2 = 40 + 42.25

Reorder the terms:
42.25 + 13w + w2 = 40 + 42.25

Combine like terms: 40 + 42.25 = 82.25
42.25 + 13w + w2 = 82.25

Factor a perfect square on the left side:
(w + 6.5)(w + 6.5) = 82.25

Calculate the square root of the right side: 9.069178574

Break this problem into two subproblems by setting 
(w + 6.5) equal to 9.069178574 and -9.069178574.

Subproblem 1

w + 6.5 = 9.069178574 Simplifying w + 6.5 = 9.069178574 Reorder the terms: 6.5 + w = 9.069178574 Solving 6.5 + w = 9.069178574 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + w = 9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + w = 9.069178574 + -6.5 w = 9.069178574 + -6.5 Combine like terms: 9.069178574 + -6.5 = 2.569178574 w = 2.569178574 Simplifying w = 2.569178574

Subproblem 2

w + 6.5 = -9.069178574 Simplifying w + 6.5 = -9.069178574 Reorder the terms: 6.5 + w = -9.069178574 Solving 6.5 + w = -9.069178574 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-6.5' to each side of the equation. 6.5 + -6.5 + w = -9.069178574 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + w = -9.069178574 + -6.5 w = -9.069178574 + -6.5 Combine like terms: -9.069178574 + -6.5 = -15.569178574 w = -15.569178574 Simplifying w = -15.569178574

Solution

The solution to the problem is based on the solutions from the subproblems. w = {2.569178574, -15.569178574}

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