121w^2-22w=1

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Solution for 121w^2-22w=1 equation:


Simplifying
121w2 + -22w = 1

Reorder the terms:
-22w + 121w2 = 1

Solving
-22w + 121w2 = 1

Solving for variable 'w'.

Reorder the terms:
-1 + -22w + 121w2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -22w + 121w2 = 0

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

Divide each side by '121'.
-0.00826446281 + -0.1818181818w + w2 = 0

Move the constant term to the right:

Add '0.00826446281' to each side of the equation.
-0.00826446281 + -0.1818181818w + 0.00826446281 + w2 = 0 + 0.00826446281

Reorder the terms:
-0.00826446281 + 0.00826446281 + -0.1818181818w + w2 = 0 + 0.00826446281

Combine like terms: -0.00826446281 + 0.00826446281 = 0.00000000000
0.00000000000 + -0.1818181818w + w2 = 0 + 0.00826446281
-0.1818181818w + w2 = 0 + 0.00826446281

Combine like terms: 0 + 0.00826446281 = 0.00826446281
-0.1818181818w + w2 = 0.00826446281

The w term is -0.1818181818w.  Take half its coefficient (-0.0909090909).
Square it (0.008264462808) and add it to both sides.

Add '0.008264462808' to each side of the equation.
-0.1818181818w + 0.008264462808 + w2 = 0.00826446281 + 0.008264462808

Reorder the terms:
0.008264462808 + -0.1818181818w + w2 = 0.00826446281 + 0.008264462808

Combine like terms: 0.00826446281 + 0.008264462808 = 0.016528925618
0.008264462808 + -0.1818181818w + w2 = 0.016528925618

Factor a perfect square on the left side:
(w + -0.0909090909)(w + -0.0909090909) = 0.016528925618

Calculate the square root of the right side: 0.128564869

Break this problem into two subproblems by setting 
(w + -0.0909090909) equal to 0.128564869 and -0.128564869.

Subproblem 1

w + -0.0909090909 = 0.128564869 Simplifying w + -0.0909090909 = 0.128564869 Reorder the terms: -0.0909090909 + w = 0.128564869 Solving -0.0909090909 + w = 0.128564869 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0909090909' to each side of the equation. -0.0909090909 + 0.0909090909 + w = 0.128564869 + 0.0909090909 Combine like terms: -0.0909090909 + 0.0909090909 = 0.0000000000 0.0000000000 + w = 0.128564869 + 0.0909090909 w = 0.128564869 + 0.0909090909 Combine like terms: 0.128564869 + 0.0909090909 = 0.2194739599 w = 0.2194739599 Simplifying w = 0.2194739599

Subproblem 2

w + -0.0909090909 = -0.128564869 Simplifying w + -0.0909090909 = -0.128564869 Reorder the terms: -0.0909090909 + w = -0.128564869 Solving -0.0909090909 + w = -0.128564869 Solving for variable 'w'. Move all terms containing w to the left, all other terms to the right. Add '0.0909090909' to each side of the equation. -0.0909090909 + 0.0909090909 + w = -0.128564869 + 0.0909090909 Combine like terms: -0.0909090909 + 0.0909090909 = 0.0000000000 0.0000000000 + w = -0.128564869 + 0.0909090909 w = -0.128564869 + 0.0909090909 Combine like terms: -0.128564869 + 0.0909090909 = -0.0376557781 w = -0.0376557781 Simplifying w = -0.0376557781

Solution

The solution to the problem is based on the solutions from the subproblems. w = {0.2194739599, -0.0376557781}

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