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