784=2w^2+w

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Solution for 784=2w^2+w equation:


Simplifying
784 = 2w2 + w

Reorder the terms:
784 = w + 2w2

Solving
784 = w + 2w2

Solving for variable 'w'.

Reorder the terms:
784 + -1w + -2w2 = w + -1w + 2w2 + -2w2

Combine like terms: w + -1w = 0
784 + -1w + -2w2 = 0 + 2w2 + -2w2
784 + -1w + -2w2 = 2w2 + -2w2

Combine like terms: 2w2 + -2w2 = 0
784 + -1w + -2w2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-392 + 0.5w + w2 = 0

Move the constant term to the right:

Add '392' to each side of the equation.
-392 + 0.5w + 392 + w2 = 0 + 392

Reorder the terms:
-392 + 392 + 0.5w + w2 = 0 + 392

Combine like terms: -392 + 392 = 0
0 + 0.5w + w2 = 0 + 392
0.5w + w2 = 0 + 392

Combine like terms: 0 + 392 = 392
0.5w + w2 = 392

The w term is 0.5w.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5w + 0.0625 + w2 = 392 + 0.0625

Reorder the terms:
0.0625 + 0.5w + w2 = 392 + 0.0625

Combine like terms: 392 + 0.0625 = 392.0625
0.0625 + 0.5w + w2 = 392.0625

Factor a perfect square on the left side:
(w + 0.25)(w + 0.25) = 392.0625

Calculate the square root of the right side: 19.800568174

Break this problem into two subproblems by setting 
(w + 0.25) equal to 19.800568174 and -19.800568174.

Subproblem 1

w + 0.25 = 19.800568174 Simplifying w + 0.25 = 19.800568174 Reorder the terms: 0.25 + w = 19.800568174 Solving 0.25 + w = 19.800568174 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = 19.800568174 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = 19.800568174 + -0.25 w = 19.800568174 + -0.25 Combine like terms: 19.800568174 + -0.25 = 19.550568174 w = 19.550568174 Simplifying w = 19.550568174

Subproblem 2

w + 0.25 = -19.800568174 Simplifying w + 0.25 = -19.800568174 Reorder the terms: 0.25 + w = -19.800568174 Solving 0.25 + w = -19.800568174 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + w = -19.800568174 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + w = -19.800568174 + -0.25 w = -19.800568174 + -0.25 Combine like terms: -19.800568174 + -0.25 = -20.050568174 w = -20.050568174 Simplifying w = -20.050568174

Solution

The solution to the problem is based on the solutions from the subproblems. w = {19.550568174, -20.050568174}

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