4w(w+5)=7

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Solution for 4w(w+5)=7 equation:


Simplifying
4w(w + 5) = 7

Reorder the terms:
4w(5 + w) = 7
(5 * 4w + w * 4w) = 7
(20w + 4w2) = 7

Solving
20w + 4w2 = 7

Solving for variable 'w'.

Reorder the terms:
-7 + 20w + 4w2 = 7 + -7

Combine like terms: 7 + -7 = 0
-7 + 20w + 4w2 = 0

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

Divide each side by '4'.
-1.75 + 5w + w2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + 5w + 1.75 + w2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + 5w + w2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + 5w + w2 = 0 + 1.75
5w + w2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
5w + w2 = 1.75

The w term is 5w.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5w + 6.25 + w2 = 1.75 + 6.25

Reorder the terms:
6.25 + 5w + w2 = 1.75 + 6.25

Combine like terms: 1.75 + 6.25 = 8
6.25 + 5w + w2 = 8

Factor a perfect square on the left side:
(w + 2.5)(w + 2.5) = 8

Calculate the square root of the right side: 2.828427125

Break this problem into two subproblems by setting 
(w + 2.5) equal to 2.828427125 and -2.828427125.

Subproblem 1

w + 2.5 = 2.828427125 Simplifying w + 2.5 = 2.828427125 Reorder the terms: 2.5 + w = 2.828427125 Solving 2.5 + w = 2.828427125 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = 2.828427125 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = 2.828427125 + -2.5 w = 2.828427125 + -2.5 Combine like terms: 2.828427125 + -2.5 = 0.328427125 w = 0.328427125 Simplifying w = 0.328427125

Subproblem 2

w + 2.5 = -2.828427125 Simplifying w + 2.5 = -2.828427125 Reorder the terms: 2.5 + w = -2.828427125 Solving 2.5 + w = -2.828427125 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + w = -2.828427125 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + w = -2.828427125 + -2.5 w = -2.828427125 + -2.5 Combine like terms: -2.828427125 + -2.5 = -5.328427125 w = -5.328427125 Simplifying w = -5.328427125

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.328427125, -5.328427125}

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