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Simplifying 2(2.5b + -9) * 6b = -7 Reorder the terms: 2(-9 + 2.5b) * 6b = -7 Reorder the terms for easier multiplication: 2 * 6b(-9 + 2.5b) = -7 Multiply 2 * 6 12b(-9 + 2.5b) = -7 (-9 * 12b + 2.5b * 12b) = -7 (-108b + 30b2) = -7 Solving -108b + 30b2 = -7 Solving for variable 'b'. Reorder the terms: 7 + -108b + 30b2 = -7 + 7 Combine like terms: -7 + 7 = 0 7 + -108b + 30b2 = 0 Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. 0.2333333333 + -3.6b + b2 = 0.0 Move the constant term to the right: Add '-0.2333333333' to each side of the equation. 0.2333333333 + -3.6b + -0.2333333333 + b2 = 0.0 + -0.2333333333 Reorder the terms: 0.2333333333 + -0.2333333333 + -3.6b + b2 = 0.0 + -0.2333333333 Combine like terms: 0.2333333333 + -0.2333333333 = 0.0000000000 0.0000000000 + -3.6b + b2 = 0.0 + -0.2333333333 -3.6b + b2 = 0.0 + -0.2333333333 Combine like terms: 0.0 + -0.2333333333 = -0.2333333333 -3.6b + b2 = -0.2333333333 The b term is -3.6b. Take half its coefficient (-1.8). Square it (3.24) and add it to both sides. Add '3.24' to each side of the equation. -3.6b + 3.24 + b2 = -0.2333333333 + 3.24 Reorder the terms: 3.24 + -3.6b + b2 = -0.2333333333 + 3.24 Combine like terms: -0.2333333333 + 3.24 = 3.0066666667 3.24 + -3.6b + b2 = 3.0066666667 Factor a perfect square on the left side: (b + -1.8)(b + -1.8) = 3.0066666667 Calculate the square root of the right side: 1.73397424 Break this problem into two subproblems by setting (b + -1.8) equal to 1.73397424 and -1.73397424.Subproblem 1
b + -1.8 = 1.73397424 Simplifying b + -1.8 = 1.73397424 Reorder the terms: -1.8 + b = 1.73397424 Solving -1.8 + b = 1.73397424 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + b = 1.73397424 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + b = 1.73397424 + 1.8 b = 1.73397424 + 1.8 Combine like terms: 1.73397424 + 1.8 = 3.53397424 b = 3.53397424 Simplifying b = 3.53397424Subproblem 2
b + -1.8 = -1.73397424 Simplifying b + -1.8 = -1.73397424 Reorder the terms: -1.8 + b = -1.73397424 Solving -1.8 + b = -1.73397424 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.8' to each side of the equation. -1.8 + 1.8 + b = -1.73397424 + 1.8 Combine like terms: -1.8 + 1.8 = 0.0 0.0 + b = -1.73397424 + 1.8 b = -1.73397424 + 1.8 Combine like terms: -1.73397424 + 1.8 = 0.06602576 b = 0.06602576 Simplifying b = 0.06602576Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.53397424, 0.06602576}
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