41w^2+w=30

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Solution for 41w^2+w=30 equation:


Simplifying
41w2 + w = 30

Reorder the terms:
w + 41w2 = 30

Solving
w + 41w2 = 30

Solving for variable 'w'.

Reorder the terms:
-30 + w + 41w2 = 30 + -30

Combine like terms: 30 + -30 = 0
-30 + w + 41w2 = 0

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

Divide each side by '41'.
-0.7317073171 + 0.02439024390w + w2 = 0

Move the constant term to the right:

Add '0.7317073171' to each side of the equation.
-0.7317073171 + 0.02439024390w + 0.7317073171 + w2 = 0 + 0.7317073171

Reorder the terms:
-0.7317073171 + 0.7317073171 + 0.02439024390w + w2 = 0 + 0.7317073171

Combine like terms: -0.7317073171 + 0.7317073171 = 0.0000000000
0.0000000000 + 0.02439024390w + w2 = 0 + 0.7317073171
0.02439024390w + w2 = 0 + 0.7317073171

Combine like terms: 0 + 0.7317073171 = 0.7317073171
0.02439024390w + w2 = 0.7317073171

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

Add '0.25' to each side of the equation.
0.02439024390w + 0.25 + w2 = 0.7317073171 + 0.25

Reorder the terms:
0.25 + 0.02439024390w + w2 = 0.7317073171 + 0.25

Combine like terms: 0.7317073171 + 0.25 = 0.9817073171
0.25 + 0.02439024390w + w2 = 0.9817073171

Factor a perfect square on the left side:
(w + 0.5)(w + 0.5) = 0.9817073171

Calculate the square root of the right side: 0.990811444

Break this problem into two subproblems by setting 
(w + 0.5) equal to 0.990811444 and -0.990811444.

Subproblem 1

w + 0.5 = 0.990811444 Simplifying w + 0.5 = 0.990811444 Reorder the terms: 0.5 + w = 0.990811444 Solving 0.5 + w = 0.990811444 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = 0.990811444 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = 0.990811444 + -0.5 w = 0.990811444 + -0.5 Combine like terms: 0.990811444 + -0.5 = 0.490811444 w = 0.490811444 Simplifying w = 0.490811444

Subproblem 2

w + 0.5 = -0.990811444 Simplifying w + 0.5 = -0.990811444 Reorder the terms: 0.5 + w = -0.990811444 Solving 0.5 + w = -0.990811444 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + w = -0.990811444 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + w = -0.990811444 + -0.5 w = -0.990811444 + -0.5 Combine like terms: -0.990811444 + -0.5 = -1.490811444 w = -1.490811444 Simplifying w = -1.490811444

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.490811444, -1.490811444}

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