61b^2-24b-21=0

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Solution for 61b^2-24b-21=0 equation:


Simplifying
61b2 + -24b + -21 = 0

Reorder the terms:
-21 + -24b + 61b2 = 0

Solving
-21 + -24b + 61b2 = 0

Solving for variable 'b'.

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

Divide each side by '61'.
-0.3442622951 + -0.393442623b + b2 = 0

Move the constant term to the right:

Add '0.3442622951' to each side of the equation.
-0.3442622951 + -0.393442623b + 0.3442622951 + b2 = 0 + 0.3442622951

Reorder the terms:
-0.3442622951 + 0.3442622951 + -0.393442623b + b2 = 0 + 0.3442622951

Combine like terms: -0.3442622951 + 0.3442622951 = 0.0000000000
0.0000000000 + -0.393442623b + b2 = 0 + 0.3442622951
-0.393442623b + b2 = 0 + 0.3442622951

Combine like terms: 0 + 0.3442622951 = 0.3442622951
-0.393442623b + b2 = 0.3442622951

The b term is -0.393442623b.  Take half its coefficient (-0.1967213115).
Square it (0.03869927440) and add it to both sides.

Add '0.03869927440' to each side of the equation.
-0.393442623b + 0.03869927440 + b2 = 0.3442622951 + 0.03869927440

Reorder the terms:
0.03869927440 + -0.393442623b + b2 = 0.3442622951 + 0.03869927440

Combine like terms: 0.3442622951 + 0.03869927440 = 0.3829615695
0.03869927440 + -0.393442623b + b2 = 0.3829615695

Factor a perfect square on the left side:
(b + -0.1967213115)(b + -0.1967213115) = 0.3829615695

Calculate the square root of the right side: 0.618838888

Break this problem into two subproblems by setting 
(b + -0.1967213115) equal to 0.618838888 and -0.618838888.

Subproblem 1

b + -0.1967213115 = 0.618838888 Simplifying b + -0.1967213115 = 0.618838888 Reorder the terms: -0.1967213115 + b = 0.618838888 Solving -0.1967213115 + b = 0.618838888 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.1967213115' to each side of the equation. -0.1967213115 + 0.1967213115 + b = 0.618838888 + 0.1967213115 Combine like terms: -0.1967213115 + 0.1967213115 = 0.0000000000 0.0000000000 + b = 0.618838888 + 0.1967213115 b = 0.618838888 + 0.1967213115 Combine like terms: 0.618838888 + 0.1967213115 = 0.8155601995 b = 0.8155601995 Simplifying b = 0.8155601995

Subproblem 2

b + -0.1967213115 = -0.618838888 Simplifying b + -0.1967213115 = -0.618838888 Reorder the terms: -0.1967213115 + b = -0.618838888 Solving -0.1967213115 + b = -0.618838888 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.1967213115' to each side of the equation. -0.1967213115 + 0.1967213115 + b = -0.618838888 + 0.1967213115 Combine like terms: -0.1967213115 + 0.1967213115 = 0.0000000000 0.0000000000 + b = -0.618838888 + 0.1967213115 b = -0.618838888 + 0.1967213115 Combine like terms: -0.618838888 + 0.1967213115 = -0.4221175765 b = -0.4221175765 Simplifying b = -0.4221175765

Solution

The solution to the problem is based on the solutions from the subproblems. b = {0.8155601995, -0.4221175765}

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