4w^2-31w-21=0

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Solution for 4w^2-31w-21=0 equation:


Simplifying
4w2 + -31w + -21 = 0

Reorder the terms:
-21 + -31w + 4w2 = 0

Solving
-21 + -31w + 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'.
-5.25 + -7.75w + w2 = 0

Move the constant term to the right:

Add '5.25' to each side of the equation.
-5.25 + -7.75w + 5.25 + w2 = 0 + 5.25

Reorder the terms:
-5.25 + 5.25 + -7.75w + w2 = 0 + 5.25

Combine like terms: -5.25 + 5.25 = 0.00
0.00 + -7.75w + w2 = 0 + 5.25
-7.75w + w2 = 0 + 5.25

Combine like terms: 0 + 5.25 = 5.25
-7.75w + w2 = 5.25

The w term is -7.75w.  Take half its coefficient (-3.875).
Square it (15.015625) and add it to both sides.

Add '15.015625' to each side of the equation.
-7.75w + 15.015625 + w2 = 5.25 + 15.015625

Reorder the terms:
15.015625 + -7.75w + w2 = 5.25 + 15.015625

Combine like terms: 5.25 + 15.015625 = 20.265625
15.015625 + -7.75w + w2 = 20.265625

Factor a perfect square on the left side:
(w + -3.875)(w + -3.875) = 20.265625

Calculate the square root of the right side: 4.501735776

Break this problem into two subproblems by setting 
(w + -3.875) equal to 4.501735776 and -4.501735776.

Subproblem 1

w + -3.875 = 4.501735776 Simplifying w + -3.875 = 4.501735776 Reorder the terms: -3.875 + w = 4.501735776 Solving -3.875 + w = 4.501735776 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + w = 4.501735776 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + w = 4.501735776 + 3.875 w = 4.501735776 + 3.875 Combine like terms: 4.501735776 + 3.875 = 8.376735776 w = 8.376735776 Simplifying w = 8.376735776

Subproblem 2

w + -3.875 = -4.501735776 Simplifying w + -3.875 = -4.501735776 Reorder the terms: -3.875 + w = -4.501735776 Solving -3.875 + w = -4.501735776 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '3.875' to each side of the equation. -3.875 + 3.875 + w = -4.501735776 + 3.875 Combine like terms: -3.875 + 3.875 = 0.000 0.000 + w = -4.501735776 + 3.875 w = -4.501735776 + 3.875 Combine like terms: -4.501735776 + 3.875 = -0.626735776 w = -0.626735776 Simplifying w = -0.626735776

Solution

The solution to the problem is based on the solutions from the subproblems. w = {8.376735776, -0.626735776}

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