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Simplifying 1.5(b + 2) + -15 = (2b + -9) * b Reorder the terms: 1.5(2 + b) + -15 = (2b + -9) * b (2 * 1.5 + b * 1.5) + -15 = (2b + -9) * b (3 + 1.5b) + -15 = (2b + -9) * b Reorder the terms: 3 + -15 + 1.5b = (2b + -9) * b Combine like terms: 3 + -15 = -12 -12 + 1.5b = (2b + -9) * b Reorder the terms: -12 + 1.5b = (-9 + 2b) * b Reorder the terms for easier multiplication: -12 + 1.5b = b(-9 + 2b) -12 + 1.5b = (-9 * b + 2b * b) -12 + 1.5b = (-9b + 2b2) Solving -12 + 1.5b = -9b + 2b2 Solving for variable 'b'. Combine like terms: 1.5b + 9b = 10.5b -12 + 10.5b + -2b2 = -9b + 2b2 + 9b + -2b2 Reorder the terms: -12 + 10.5b + -2b2 = -9b + 9b + 2b2 + -2b2 Combine like terms: -9b + 9b = 0 -12 + 10.5b + -2b2 = 0 + 2b2 + -2b2 -12 + 10.5b + -2b2 = 2b2 + -2b2 Combine like terms: 2b2 + -2b2 = 0 -12 + 10.5b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 6 + -5.25b + b2 = 0 Move the constant term to the right: Add '-6' to each side of the equation. 6 + -5.25b + -6 + b2 = 0 + -6 Reorder the terms: 6 + -6 + -5.25b + b2 = 0 + -6 Combine like terms: 6 + -6 = 0 0 + -5.25b + b2 = 0 + -6 -5.25b + b2 = 0 + -6 Combine like terms: 0 + -6 = -6 -5.25b + b2 = -6 The b term is -5.25b. Take half its coefficient (-2.625). Square it (6.890625) and add it to both sides. Add '6.890625' to each side of the equation. -5.25b + 6.890625 + b2 = -6 + 6.890625 Reorder the terms: 6.890625 + -5.25b + b2 = -6 + 6.890625 Combine like terms: -6 + 6.890625 = 0.890625 6.890625 + -5.25b + b2 = 0.890625 Factor a perfect square on the left side: (b + -2.625)(b + -2.625) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (b + -2.625) equal to 0.943729304 and -0.943729304.Subproblem 1
b + -2.625 = 0.943729304 Simplifying b + -2.625 = 0.943729304 Reorder the terms: -2.625 + b = 0.943729304 Solving -2.625 + b = 0.943729304 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + b = 0.943729304 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + b = 0.943729304 + 2.625 b = 0.943729304 + 2.625 Combine like terms: 0.943729304 + 2.625 = 3.568729304 b = 3.568729304 Simplifying b = 3.568729304Subproblem 2
b + -2.625 = -0.943729304 Simplifying b + -2.625 = -0.943729304 Reorder the terms: -2.625 + b = -0.943729304 Solving -2.625 + b = -0.943729304 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.625' to each side of the equation. -2.625 + 2.625 + b = -0.943729304 + 2.625 Combine like terms: -2.625 + 2.625 = 0.000 0.000 + b = -0.943729304 + 2.625 b = -0.943729304 + 2.625 Combine like terms: -0.943729304 + 2.625 = 1.681270696 b = 1.681270696 Simplifying b = 1.681270696Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.568729304, 1.681270696}
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