5(4w+4)2w=7(w+8)-6

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Solution for 5(4w+4)2w=7(w+8)-6 equation:


Simplifying
5(4w + 4) * 2w = 7(w + 8) + -6

Reorder the terms:
5(4 + 4w) * 2w = 7(w + 8) + -6

Reorder the terms for easier multiplication:
5 * 2w(4 + 4w) = 7(w + 8) + -6

Multiply 5 * 2
10w(4 + 4w) = 7(w + 8) + -6
(4 * 10w + 4w * 10w) = 7(w + 8) + -6
(40w + 40w2) = 7(w + 8) + -6

Reorder the terms:
40w + 40w2 = 7(8 + w) + -6
40w + 40w2 = (8 * 7 + w * 7) + -6
40w + 40w2 = (56 + 7w) + -6

Reorder the terms:
40w + 40w2 = 56 + -6 + 7w

Combine like terms: 56 + -6 = 50
40w + 40w2 = 50 + 7w

Solving
40w + 40w2 = 50 + 7w

Solving for variable 'w'.

Reorder the terms:
-50 + 40w + -7w + 40w2 = 50 + 7w + -50 + -7w

Combine like terms: 40w + -7w = 33w
-50 + 33w + 40w2 = 50 + 7w + -50 + -7w

Reorder the terms:
-50 + 33w + 40w2 = 50 + -50 + 7w + -7w

Combine like terms: 50 + -50 = 0
-50 + 33w + 40w2 = 0 + 7w + -7w
-50 + 33w + 40w2 = 7w + -7w

Combine like terms: 7w + -7w = 0
-50 + 33w + 40w2 = 0

Begin completing the square.  Divide all terms by
40 the coefficient of the squared term: 

Divide each side by '40'.
-1.25 + 0.825w + w2 = 0.0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + 0.825w + 1.25 + w2 = 0.0 + 1.25

Reorder the terms:
-1.25 + 1.25 + 0.825w + w2 = 0.0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + 0.825w + w2 = 0.0 + 1.25
0.825w + w2 = 0.0 + 1.25

Combine like terms: 0.0 + 1.25 = 1.25
0.825w + w2 = 1.25

The w term is 0.825w.  Take half its coefficient (0.4125).
Square it (0.17015625) and add it to both sides.

Add '0.17015625' to each side of the equation.
0.825w + 0.17015625 + w2 = 1.25 + 0.17015625

Reorder the terms:
0.17015625 + 0.825w + w2 = 1.25 + 0.17015625

Combine like terms: 1.25 + 0.17015625 = 1.42015625
0.17015625 + 0.825w + w2 = 1.42015625

Factor a perfect square on the left side:
(w + 0.4125)(w + 0.4125) = 1.42015625

Calculate the square root of the right side: 1.191703088

Break this problem into two subproblems by setting 
(w + 0.4125) equal to 1.191703088 and -1.191703088.

Subproblem 1

w + 0.4125 = 1.191703088 Simplifying w + 0.4125 = 1.191703088 Reorder the terms: 0.4125 + w = 1.191703088 Solving 0.4125 + w = 1.191703088 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.4125' to each side of the equation. 0.4125 + -0.4125 + w = 1.191703088 + -0.4125 Combine like terms: 0.4125 + -0.4125 = 0.0000 0.0000 + w = 1.191703088 + -0.4125 w = 1.191703088 + -0.4125 Combine like terms: 1.191703088 + -0.4125 = 0.779203088 w = 0.779203088 Simplifying w = 0.779203088

Subproblem 2

w + 0.4125 = -1.191703088 Simplifying w + 0.4125 = -1.191703088 Reorder the terms: 0.4125 + w = -1.191703088 Solving 0.4125 + w = -1.191703088 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.4125' to each side of the equation. 0.4125 + -0.4125 + w = -1.191703088 + -0.4125 Combine like terms: 0.4125 + -0.4125 = 0.0000 0.0000 + w = -1.191703088 + -0.4125 w = -1.191703088 + -0.4125 Combine like terms: -1.191703088 + -0.4125 = -1.604203088 w = -1.604203088 Simplifying w = -1.604203088

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.779203088, -1.604203088}

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