4w^2-21w-5=0

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Solution for 4w^2-21w-5=0 equation:


Simplifying
4w2 + -21w + -5 = 0

Reorder the terms:
-5 + -21w + 4w2 = 0

Solving
-5 + -21w + 4w2 = 0

Solving for variable 'w'.

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

Divide each side by '4'.
-1.25 + -5.25w + w2 = 0

Move the constant term to the right:

Add '1.25' to each side of the equation.
-1.25 + -5.25w + 1.25 + w2 = 0 + 1.25

Reorder the terms:
-1.25 + 1.25 + -5.25w + w2 = 0 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + -5.25w + w2 = 0 + 1.25
-5.25w + w2 = 0 + 1.25

Combine like terms: 0 + 1.25 = 1.25
-5.25w + w2 = 1.25

The w term is -5.25w.  Take half its coefficient (-2.625).
Square it (6.890625) and add it to both sides.

Add '6.890625' to each side of the equation.
-5.25w + 6.890625 + w2 = 1.25 + 6.890625

Reorder the terms:
6.890625 + -5.25w + w2 = 1.25 + 6.890625

Combine like terms: 1.25 + 6.890625 = 8.140625
6.890625 + -5.25w + w2 = 8.140625

Factor a perfect square on the left side:
(w + -2.625)(w + -2.625) = 8.140625

Calculate the square root of the right side: 2.853178053

Break this problem into two subproblems by setting 
(w + -2.625) equal to 2.853178053 and -2.853178053.

Subproblem 1

w + -2.625 = 2.853178053 Simplifying w + -2.625 = 2.853178053 Reorder the terms: -2.625 + w = 2.853178053 Solving -2.625 + w = 2.853178053 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + w = 2.853178053 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + w = 2.853178053 + 2.625 w = 2.853178053 + 2.625 Combine like terms: 2.853178053 + 2.625 = 5.478178053 w = 5.478178053 Simplifying w = 5.478178053

Subproblem 2

w + -2.625 = -2.853178053 Simplifying w + -2.625 = -2.853178053 Reorder the terms: -2.625 + w = -2.853178053 Solving -2.625 + w = -2.853178053 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + w = -2.853178053 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + w = -2.853178053 + 2.625 w = -2.853178053 + 2.625 Combine like terms: -2.853178053 + 2.625 = -0.228178053 w = -0.228178053 Simplifying w = -0.228178053

Solution

The solution to the problem is based on the solutions from the subproblems. w = {5.478178053, -0.228178053}

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