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Simplifying (2b + -3)(3b + -8) = 8 Reorder the terms: (-3 + 2b)(3b + -8) = 8 Reorder the terms: (-3 + 2b)(-8 + 3b) = 8 Multiply (-3 + 2b) * (-8 + 3b) (-3(-8 + 3b) + 2b * (-8 + 3b)) = 8 ((-8 * -3 + 3b * -3) + 2b * (-8 + 3b)) = 8 ((24 + -9b) + 2b * (-8 + 3b)) = 8 (24 + -9b + (-8 * 2b + 3b * 2b)) = 8 (24 + -9b + (-16b + 6b2)) = 8 Combine like terms: -9b + -16b = -25b (24 + -25b + 6b2) = 8 Solving 24 + -25b + 6b2 = 8 Solving for variable 'b'. Reorder the terms: 24 + -8 + -25b + 6b2 = 8 + -8 Combine like terms: 24 + -8 = 16 16 + -25b + 6b2 = 8 + -8 Combine like terms: 8 + -8 = 0 16 + -25b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 2.666666667 + -4.166666667b + b2 = 0 Move the constant term to the right: Add '-2.666666667' to each side of the equation. 2.666666667 + -4.166666667b + -2.666666667 + b2 = 0 + -2.666666667 Reorder the terms: 2.666666667 + -2.666666667 + -4.166666667b + b2 = 0 + -2.666666667 Combine like terms: 2.666666667 + -2.666666667 = 0.000000000 0.000000000 + -4.166666667b + b2 = 0 + -2.666666667 -4.166666667b + b2 = 0 + -2.666666667 Combine like terms: 0 + -2.666666667 = -2.666666667 -4.166666667b + b2 = -2.666666667 The b term is -4.166666667b. Take half its coefficient (-2.083333334). Square it (4.340277781) and add it to both sides. Add '4.340277781' to each side of the equation. -4.166666667b + 4.340277781 + b2 = -2.666666667 + 4.340277781 Reorder the terms: 4.340277781 + -4.166666667b + b2 = -2.666666667 + 4.340277781 Combine like terms: -2.666666667 + 4.340277781 = 1.673611114 4.340277781 + -4.166666667b + b2 = 1.673611114 Factor a perfect square on the left side: (b + -2.083333334)(b + -2.083333334) = 1.673611114 Calculate the square root of the right side: 1.293681226 Break this problem into two subproblems by setting (b + -2.083333334) equal to 1.293681226 and -1.293681226.Subproblem 1
b + -2.083333334 = 1.293681226 Simplifying b + -2.083333334 = 1.293681226 Reorder the terms: -2.083333334 + b = 1.293681226 Solving -2.083333334 + b = 1.293681226 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.083333334' to each side of the equation. -2.083333334 + 2.083333334 + b = 1.293681226 + 2.083333334 Combine like terms: -2.083333334 + 2.083333334 = 0.000000000 0.000000000 + b = 1.293681226 + 2.083333334 b = 1.293681226 + 2.083333334 Combine like terms: 1.293681226 + 2.083333334 = 3.37701456 b = 3.37701456 Simplifying b = 3.37701456Subproblem 2
b + -2.083333334 = -1.293681226 Simplifying b + -2.083333334 = -1.293681226 Reorder the terms: -2.083333334 + b = -1.293681226 Solving -2.083333334 + b = -1.293681226 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.083333334' to each side of the equation. -2.083333334 + 2.083333334 + b = -1.293681226 + 2.083333334 Combine like terms: -2.083333334 + 2.083333334 = 0.000000000 0.000000000 + b = -1.293681226 + 2.083333334 b = -1.293681226 + 2.083333334 Combine like terms: -1.293681226 + 2.083333334 = 0.789652108 b = 0.789652108 Simplifying b = 0.789652108Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.37701456, 0.789652108}
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