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Simplifying 0 = 13b2 + -60b + 21 Reorder the terms: 0 = 21 + -60b + 13b2 Solving 0 = 21 + -60b + 13b2 Solving for variable 'b'. Combine like terms: 0 + -21 = -21 -21 + 60b + -13b2 = 21 + -60b + 13b2 + -21 + 60b + -13b2 Reorder the terms: -21 + 60b + -13b2 = 21 + -21 + -60b + 60b + 13b2 + -13b2 Combine like terms: 21 + -21 = 0 -21 + 60b + -13b2 = 0 + -60b + 60b + 13b2 + -13b2 -21 + 60b + -13b2 = -60b + 60b + 13b2 + -13b2 Combine like terms: -60b + 60b = 0 -21 + 60b + -13b2 = 0 + 13b2 + -13b2 -21 + 60b + -13b2 = 13b2 + -13b2 Combine like terms: 13b2 + -13b2 = 0 -21 + 60b + -13b2 = 0 Begin completing the square. Divide all terms by -13 the coefficient of the squared term: Divide each side by '-13'. 1.615384615 + -4.615384615b + b2 = 0 Move the constant term to the right: Add '-1.615384615' to each side of the equation. 1.615384615 + -4.615384615b + -1.615384615 + b2 = 0 + -1.615384615 Reorder the terms: 1.615384615 + -1.615384615 + -4.615384615b + b2 = 0 + -1.615384615 Combine like terms: 1.615384615 + -1.615384615 = 0.000000000 0.000000000 + -4.615384615b + b2 = 0 + -1.615384615 -4.615384615b + b2 = 0 + -1.615384615 Combine like terms: 0 + -1.615384615 = -1.615384615 -4.615384615b + b2 = -1.615384615 The b term is -4.615384615b. Take half its coefficient (-2.307692308). Square it (5.325443788) and add it to both sides. Add '5.325443788' to each side of the equation. -4.615384615b + 5.325443788 + b2 = -1.615384615 + 5.325443788 Reorder the terms: 5.325443788 + -4.615384615b + b2 = -1.615384615 + 5.325443788 Combine like terms: -1.615384615 + 5.325443788 = 3.710059173 5.325443788 + -4.615384615b + b2 = 3.710059173 Factor a perfect square on the left side: (b + -2.307692308)(b + -2.307692308) = 3.710059173 Calculate the square root of the right side: 1.926151389 Break this problem into two subproblems by setting (b + -2.307692308) equal to 1.926151389 and -1.926151389.Subproblem 1
b + -2.307692308 = 1.926151389 Simplifying b + -2.307692308 = 1.926151389 Reorder the terms: -2.307692308 + b = 1.926151389 Solving -2.307692308 + b = 1.926151389 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + b = 1.926151389 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + b = 1.926151389 + 2.307692308 b = 1.926151389 + 2.307692308 Combine like terms: 1.926151389 + 2.307692308 = 4.233843697 b = 4.233843697 Simplifying b = 4.233843697Subproblem 2
b + -2.307692308 = -1.926151389 Simplifying b + -2.307692308 = -1.926151389 Reorder the terms: -2.307692308 + b = -1.926151389 Solving -2.307692308 + b = -1.926151389 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.307692308' to each side of the equation. -2.307692308 + 2.307692308 + b = -1.926151389 + 2.307692308 Combine like terms: -2.307692308 + 2.307692308 = 0.000000000 0.000000000 + b = -1.926151389 + 2.307692308 b = -1.926151389 + 2.307692308 Combine like terms: -1.926151389 + 2.307692308 = 0.381540919 b = 0.381540919 Simplifying b = 0.381540919Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.233843697, 0.381540919}
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