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Simplifying (1.5b + -6)(2b + 5) = 0 Reorder the terms: (-6 + 1.5b)(2b + 5) = 0 Reorder the terms: (-6 + 1.5b)(5 + 2b) = 0 Multiply (-6 + 1.5b) * (5 + 2b) (-6(5 + 2b) + 1.5b * (5 + 2b)) = 0 ((5 * -6 + 2b * -6) + 1.5b * (5 + 2b)) = 0 ((-30 + -12b) + 1.5b * (5 + 2b)) = 0 (-30 + -12b + (5 * 1.5b + 2b * 1.5b)) = 0 (-30 + -12b + (7.5b + 3b2)) = 0 Combine like terms: -12b + 7.5b = -4.5b (-30 + -4.5b + 3b2) = 0 Solving -30 + -4.5b + 3b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -10 + -1.5b + b2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + -1.5b + 10 + b2 = 0 + 10 Reorder the terms: -10 + 10 + -1.5b + b2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + -1.5b + b2 = 0 + 10 -1.5b + b2 = 0 + 10 Combine like terms: 0 + 10 = 10 -1.5b + b2 = 10 The b term is -1.5b. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5b + 0.5625 + b2 = 10 + 0.5625 Reorder the terms: 0.5625 + -1.5b + b2 = 10 + 0.5625 Combine like terms: 10 + 0.5625 = 10.5625 0.5625 + -1.5b + b2 = 10.5625 Factor a perfect square on the left side: (b + -0.75)(b + -0.75) = 10.5625 Calculate the square root of the right side: 3.25 Break this problem into two subproblems by setting (b + -0.75) equal to 3.25 and -3.25.Subproblem 1
b + -0.75 = 3.25 Simplifying b + -0.75 = 3.25 Reorder the terms: -0.75 + b = 3.25 Solving -0.75 + b = 3.25 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = 3.25 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = 3.25 + 0.75 b = 3.25 + 0.75 Combine like terms: 3.25 + 0.75 = 4 b = 4 Simplifying b = 4Subproblem 2
b + -0.75 = -3.25 Simplifying b + -0.75 = -3.25 Reorder the terms: -0.75 + b = -3.25 Solving -0.75 + b = -3.25 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = -3.25 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = -3.25 + 0.75 b = -3.25 + 0.75 Combine like terms: -3.25 + 0.75 = -2.5 b = -2.5 Simplifying b = -2.5Solution
The solution to the problem is based on the solutions from the subproblems. b = {4, -2.5}
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