20b=1+4b^2

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Solution for 20b=1+4b^2 equation:


Simplifying
20b = 1 + 4b2

Solving
20b = 1 + 4b2

Solving for variable 'b'.

Reorder the terms:
-1 + 20b + -4b2 = 1 + 4b2 + -1 + -4b2

Reorder the terms:
-1 + 20b + -4b2 = 1 + -1 + 4b2 + -4b2

Combine like terms: 1 + -1 = 0
-1 + 20b + -4b2 = 0 + 4b2 + -4b2
-1 + 20b + -4b2 = 4b2 + -4b2

Combine like terms: 4b2 + -4b2 = 0
-1 + 20b + -4b2 = 0

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

Divide each side by '-4'.
0.25 + -5b + b2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -5b + -0.25 + b2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -5b + b2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -5b + b2 = 0 + -0.25
-5b + b2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-5b + b2 = -0.25

The b term is -5b.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5b + 6.25 + b2 = -0.25 + 6.25

Reorder the terms:
6.25 + -5b + b2 = -0.25 + 6.25

Combine like terms: -0.25 + 6.25 = 6
6.25 + -5b + b2 = 6

Factor a perfect square on the left side:
(b + -2.5)(b + -2.5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(b + -2.5) equal to 2.449489743 and -2.449489743.

Subproblem 1

b + -2.5 = 2.449489743 Simplifying b + -2.5 = 2.449489743 Reorder the terms: -2.5 + b = 2.449489743 Solving -2.5 + b = 2.449489743 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + b = 2.449489743 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = 2.449489743 + 2.5 b = 2.449489743 + 2.5 Combine like terms: 2.449489743 + 2.5 = 4.949489743 b = 4.949489743 Simplifying b = 4.949489743

Subproblem 2

b + -2.5 = -2.449489743 Simplifying b + -2.5 = -2.449489743 Reorder the terms: -2.5 + b = -2.449489743 Solving -2.5 + b = -2.449489743 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + b = -2.449489743 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = -2.449489743 + 2.5 b = -2.449489743 + 2.5 Combine like terms: -2.449489743 + 2.5 = 0.050510257 b = 0.050510257 Simplifying b = 0.050510257

Solution

The solution to the problem is based on the solutions from the subproblems. b = {4.949489743, 0.050510257}

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