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Simplifying (6b2 + 5b) + (8b + -20) = 0 Reorder the terms: (5b + 6b2) + (8b + -20) = 0 Remove parenthesis around (5b + 6b2) 5b + 6b2 + (8b + -20) = 0 Reorder the terms: 5b + 6b2 + (-20 + 8b) = 0 Remove parenthesis around (-20 + 8b) 5b + 6b2 + -20 + 8b = 0 Reorder the terms: -20 + 5b + 8b + 6b2 = 0 Combine like terms: 5b + 8b = 13b -20 + 13b + 6b2 = 0 Solving -20 + 13b + 6b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.333333333 + 2.166666667b + b2 = 0 Move the constant term to the right: Add '3.333333333' to each side of the equation. -3.333333333 + 2.166666667b + 3.333333333 + b2 = 0 + 3.333333333 Reorder the terms: -3.333333333 + 3.333333333 + 2.166666667b + b2 = 0 + 3.333333333 Combine like terms: -3.333333333 + 3.333333333 = 0.000000000 0.000000000 + 2.166666667b + b2 = 0 + 3.333333333 2.166666667b + b2 = 0 + 3.333333333 Combine like terms: 0 + 3.333333333 = 3.333333333 2.166666667b + b2 = 3.333333333 The b term is 2.166666667b. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667b + 1.173611113 + b2 = 3.333333333 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667b + b2 = 3.333333333 + 1.173611113 Combine like terms: 3.333333333 + 1.173611113 = 4.506944446 1.173611113 + 2.166666667b + b2 = 4.506944446 Factor a perfect square on the left side: (b + 1.083333334)(b + 1.083333334) = 4.506944446 Calculate the square root of the right side: 2.122956534 Break this problem into two subproblems by setting (b + 1.083333334) equal to 2.122956534 and -2.122956534.Subproblem 1
b + 1.083333334 = 2.122956534 Simplifying b + 1.083333334 = 2.122956534 Reorder the terms: 1.083333334 + b = 2.122956534 Solving 1.083333334 + b = 2.122956534 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + b = 2.122956534 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + b = 2.122956534 + -1.083333334 b = 2.122956534 + -1.083333334 Combine like terms: 2.122956534 + -1.083333334 = 1.0396232 b = 1.0396232 Simplifying b = 1.0396232Subproblem 2
b + 1.083333334 = -2.122956534 Simplifying b + 1.083333334 = -2.122956534 Reorder the terms: 1.083333334 + b = -2.122956534 Solving 1.083333334 + b = -2.122956534 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + b = -2.122956534 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + b = -2.122956534 + -1.083333334 b = -2.122956534 + -1.083333334 Combine like terms: -2.122956534 + -1.083333334 = -3.206289868 b = -3.206289868 Simplifying b = -3.206289868Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.0396232, -3.206289868}
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