(w+1.5)(w+10)+1.5=399

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Solution for (w+1.5)(w+10)+1.5=399 equation:


Simplifying
(w + 1.5)(w + 10) + 1.5 = 399

Reorder the terms:
(1.5 + w)(w + 10) + 1.5 = 399

Reorder the terms:
(1.5 + w)(10 + w) + 1.5 = 399

Multiply (1.5 + w) * (10 + w)
(1.5(10 + w) + w(10 + w)) + 1.5 = 399
((10 * 1.5 + w * 1.5) + w(10 + w)) + 1.5 = 399
((15 + 1.5w) + w(10 + w)) + 1.5 = 399
(15 + 1.5w + (10 * w + w * w)) + 1.5 = 399
(15 + 1.5w + (10w + w2)) + 1.5 = 399

Combine like terms: 1.5w + 10w = 11.5w
(15 + 11.5w + w2) + 1.5 = 399

Reorder the terms:
15 + 1.5 + 11.5w + w2 = 399

Combine like terms: 15 + 1.5 = 16.5
16.5 + 11.5w + w2 = 399

Solving
16.5 + 11.5w + w2 = 399

Solving for variable 'w'.

Reorder the terms:
16.5 + -399 + 11.5w + w2 = 399 + -399

Combine like terms: 16.5 + -399 = -382.5
-382.5 + 11.5w + w2 = 399 + -399

Combine like terms: 399 + -399 = 0
-382.5 + 11.5w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '382.5' to each side of the equation.
-382.5 + 11.5w + 382.5 + w2 = 0 + 382.5

Reorder the terms:
-382.5 + 382.5 + 11.5w + w2 = 0 + 382.5

Combine like terms: -382.5 + 382.5 = 0.0
0.0 + 11.5w + w2 = 0 + 382.5
11.5w + w2 = 0 + 382.5

Combine like terms: 0 + 382.5 = 382.5
11.5w + w2 = 382.5

The w term is 11.5w.  Take half its coefficient (5.75).
Square it (33.0625) and add it to both sides.

Add '33.0625' to each side of the equation.
11.5w + 33.0625 + w2 = 382.5 + 33.0625

Reorder the terms:
33.0625 + 11.5w + w2 = 382.5 + 33.0625

Combine like terms: 382.5 + 33.0625 = 415.5625
33.0625 + 11.5w + w2 = 415.5625

Factor a perfect square on the left side:
(w + 5.75)(w + 5.75) = 415.5625

Calculate the square root of the right side: 20.385350132

Break this problem into two subproblems by setting 
(w + 5.75) equal to 20.385350132 and -20.385350132.

Subproblem 1

w + 5.75 = 20.385350132 Simplifying w + 5.75 = 20.385350132 Reorder the terms: 5.75 + w = 20.385350132 Solving 5.75 + w = 20.385350132 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + w = 20.385350132 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + w = 20.385350132 + -5.75 w = 20.385350132 + -5.75 Combine like terms: 20.385350132 + -5.75 = 14.635350132 w = 14.635350132 Simplifying w = 14.635350132

Subproblem 2

w + 5.75 = -20.385350132 Simplifying w + 5.75 = -20.385350132 Reorder the terms: 5.75 + w = -20.385350132 Solving 5.75 + w = -20.385350132 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-5.75' to each side of the equation. 5.75 + -5.75 + w = -20.385350132 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + w = -20.385350132 + -5.75 w = -20.385350132 + -5.75 Combine like terms: -20.385350132 + -5.75 = -26.135350132 w = -26.135350132 Simplifying w = -26.135350132

Solution

The solution to the problem is based on the solutions from the subproblems. w = {14.635350132, -26.135350132}

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