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Simplifying -5b + 24 = -8(b + -6) * 6b Reorder the terms: 24 + -5b = -8(b + -6) * 6b Reorder the terms: 24 + -5b = -8(-6 + b) * 6b Reorder the terms for easier multiplication: 24 + -5b = -8 * 6b(-6 + b) Multiply -8 * 6 24 + -5b = -48b(-6 + b) 24 + -5b = (-6 * -48b + b * -48b) 24 + -5b = (288b + -48b2) Solving 24 + -5b = 288b + -48b2 Solving for variable 'b'. Combine like terms: -5b + -288b = -293b 24 + -293b + 48b2 = 288b + -48b2 + -288b + 48b2 Reorder the terms: 24 + -293b + 48b2 = 288b + -288b + -48b2 + 48b2 Combine like terms: 288b + -288b = 0 24 + -293b + 48b2 = 0 + -48b2 + 48b2 24 + -293b + 48b2 = -48b2 + 48b2 Combine like terms: -48b2 + 48b2 = 0 24 + -293b + 48b2 = 0 Begin completing the square. Divide all terms by 48 the coefficient of the squared term: Divide each side by '48'. 0.5 + -6.104166667b + b2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -6.104166667b + -0.5 + b2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -6.104166667b + b2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -6.104166667b + b2 = 0 + -0.5 -6.104166667b + b2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -6.104166667b + b2 = -0.5 The b term is -6.104166667b. Take half its coefficient (-3.052083334). Square it (9.315212678) and add it to both sides. Add '9.315212678' to each side of the equation. -6.104166667b + 9.315212678 + b2 = -0.5 + 9.315212678 Reorder the terms: 9.315212678 + -6.104166667b + b2 = -0.5 + 9.315212678 Combine like terms: -0.5 + 9.315212678 = 8.815212678 9.315212678 + -6.104166667b + b2 = 8.815212678 Factor a perfect square on the left side: (b + -3.052083334)(b + -3.052083334) = 8.815212678 Calculate the square root of the right side: 2.969042384 Break this problem into two subproblems by setting (b + -3.052083334) equal to 2.969042384 and -2.969042384.Subproblem 1
b + -3.052083334 = 2.969042384 Simplifying b + -3.052083334 = 2.969042384 Reorder the terms: -3.052083334 + b = 2.969042384 Solving -3.052083334 + b = 2.969042384 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.052083334' to each side of the equation. -3.052083334 + 3.052083334 + b = 2.969042384 + 3.052083334 Combine like terms: -3.052083334 + 3.052083334 = 0.000000000 0.000000000 + b = 2.969042384 + 3.052083334 b = 2.969042384 + 3.052083334 Combine like terms: 2.969042384 + 3.052083334 = 6.021125718 b = 6.021125718 Simplifying b = 6.021125718Subproblem 2
b + -3.052083334 = -2.969042384 Simplifying b + -3.052083334 = -2.969042384 Reorder the terms: -3.052083334 + b = -2.969042384 Solving -3.052083334 + b = -2.969042384 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.052083334' to each side of the equation. -3.052083334 + 3.052083334 + b = -2.969042384 + 3.052083334 Combine like terms: -3.052083334 + 3.052083334 = 0.000000000 0.000000000 + b = -2.969042384 + 3.052083334 b = -2.969042384 + 3.052083334 Combine like terms: -2.969042384 + 3.052083334 = 0.08304095 b = 0.08304095 Simplifying b = 0.08304095Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.021125718, 0.08304095}
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