w^2+45w=1000

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Solution for w^2+45w=1000 equation:


Simplifying
w2 + 45w = 1000

Reorder the terms:
45w + w2 = 1000

Solving
45w + w2 = 1000

Solving for variable 'w'.

Reorder the terms:
-1000 + 45w + w2 = 1000 + -1000

Combine like terms: 1000 + -1000 = 0
-1000 + 45w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1000' to each side of the equation.
-1000 + 45w + 1000 + w2 = 0 + 1000

Reorder the terms:
-1000 + 1000 + 45w + w2 = 0 + 1000

Combine like terms: -1000 + 1000 = 0
0 + 45w + w2 = 0 + 1000
45w + w2 = 0 + 1000

Combine like terms: 0 + 1000 = 1000
45w + w2 = 1000

The w term is 45w.  Take half its coefficient (22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
45w + 506.25 + w2 = 1000 + 506.25

Reorder the terms:
506.25 + 45w + w2 = 1000 + 506.25

Combine like terms: 1000 + 506.25 = 1506.25
506.25 + 45w + w2 = 1506.25

Factor a perfect square on the left side:
(w + 22.5)(w + 22.5) = 1506.25

Calculate the square root of the right side: 38.810436741

Break this problem into two subproblems by setting 
(w + 22.5) equal to 38.810436741 and -38.810436741.

Subproblem 1

w + 22.5 = 38.810436741 Simplifying w + 22.5 = 38.810436741 Reorder the terms: 22.5 + w = 38.810436741 Solving 22.5 + w = 38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + w = 38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + w = 38.810436741 + -22.5 w = 38.810436741 + -22.5 Combine like terms: 38.810436741 + -22.5 = 16.310436741 w = 16.310436741 Simplifying w = 16.310436741

Subproblem 2

w + 22.5 = -38.810436741 Simplifying w + 22.5 = -38.810436741 Reorder the terms: 22.5 + w = -38.810436741 Solving 22.5 + w = -38.810436741 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-22.5' to each side of the equation. 22.5 + -22.5 + w = -38.810436741 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 0.0 + w = -38.810436741 + -22.5 w = -38.810436741 + -22.5 Combine like terms: -38.810436741 + -22.5 = -61.310436741 w = -61.310436741 Simplifying w = -61.310436741

Solution

The solution to the problem is based on the solutions from the subproblems. w = {16.310436741, -61.310436741}

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