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Simplifying 12b[5 + 2b] + -7 = 0 [5 * 12b + 2b * 12b] + -7 = 0 [60b + 24b2] + -7 = 0 Reorder the terms: -7 + 60b + 24b2 = 0 Solving -7 + 60b + 24b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.2916666667 + 2.5b + b2 = 0 Move the constant term to the right: Add '0.2916666667' to each side of the equation. -0.2916666667 + 2.5b + 0.2916666667 + b2 = 0 + 0.2916666667 Reorder the terms: -0.2916666667 + 0.2916666667 + 2.5b + b2 = 0 + 0.2916666667 Combine like terms: -0.2916666667 + 0.2916666667 = 0.0000000000 0.0000000000 + 2.5b + b2 = 0 + 0.2916666667 2.5b + b2 = 0 + 0.2916666667 Combine like terms: 0 + 0.2916666667 = 0.2916666667 2.5b + b2 = 0.2916666667 The b term is 2.5b. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5b + 1.5625 + b2 = 0.2916666667 + 1.5625 Reorder the terms: 1.5625 + 2.5b + b2 = 0.2916666667 + 1.5625 Combine like terms: 0.2916666667 + 1.5625 = 1.8541666667 1.5625 + 2.5b + b2 = 1.8541666667 Factor a perfect square on the left side: (b + 1.25)(b + 1.25) = 1.8541666667 Calculate the square root of the right side: 1.361677887 Break this problem into two subproblems by setting (b + 1.25) equal to 1.361677887 and -1.361677887.Subproblem 1
b + 1.25 = 1.361677887 Simplifying b + 1.25 = 1.361677887 Reorder the terms: 1.25 + b = 1.361677887 Solving 1.25 + b = 1.361677887 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + b = 1.361677887 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + b = 1.361677887 + -1.25 b = 1.361677887 + -1.25 Combine like terms: 1.361677887 + -1.25 = 0.111677887 b = 0.111677887 Simplifying b = 0.111677887Subproblem 2
b + 1.25 = -1.361677887 Simplifying b + 1.25 = -1.361677887 Reorder the terms: 1.25 + b = -1.361677887 Solving 1.25 + b = -1.361677887 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + b = -1.361677887 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + b = -1.361677887 + -1.25 b = -1.361677887 + -1.25 Combine like terms: -1.361677887 + -1.25 = -2.611677887 b = -2.611677887 Simplifying b = -2.611677887Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.111677887, -2.611677887}
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