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Simplifying 20(b + -15) * b = 19 Reorder the terms: 20(-15 + b) * b = 19 Reorder the terms for easier multiplication: 20b(-15 + b) = 19 (-15 * 20b + b * 20b) = 19 (-300b + 20b2) = 19 Solving -300b + 20b2 = 19 Solving for variable 'b'. Reorder the terms: -19 + -300b + 20b2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + -300b + 20b2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. -0.95 + -15b + b2 = 0 Move the constant term to the right: Add '0.95' to each side of the equation. -0.95 + -15b + 0.95 + b2 = 0 + 0.95 Reorder the terms: -0.95 + 0.95 + -15b + b2 = 0 + 0.95 Combine like terms: -0.95 + 0.95 = 0.00 0.00 + -15b + b2 = 0 + 0.95 -15b + b2 = 0 + 0.95 Combine like terms: 0 + 0.95 = 0.95 -15b + b2 = 0.95 The b term is -15b. Take half its coefficient (-7.5). Square it (56.25) and add it to both sides. Add '56.25' to each side of the equation. -15b + 56.25 + b2 = 0.95 + 56.25 Reorder the terms: 56.25 + -15b + b2 = 0.95 + 56.25 Combine like terms: 0.95 + 56.25 = 57.2 56.25 + -15b + b2 = 57.2 Factor a perfect square on the left side: (b + -7.5)(b + -7.5) = 57.2 Calculate the square root of the right side: 7.56306816 Break this problem into two subproblems by setting (b + -7.5) equal to 7.56306816 and -7.56306816.Subproblem 1
b + -7.5 = 7.56306816 Simplifying b + -7.5 = 7.56306816 Reorder the terms: -7.5 + b = 7.56306816 Solving -7.5 + b = 7.56306816 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + b = 7.56306816 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + b = 7.56306816 + 7.5 b = 7.56306816 + 7.5 Combine like terms: 7.56306816 + 7.5 = 15.06306816 b = 15.06306816 Simplifying b = 15.06306816Subproblem 2
b + -7.5 = -7.56306816 Simplifying b + -7.5 = -7.56306816 Reorder the terms: -7.5 + b = -7.56306816 Solving -7.5 + b = -7.56306816 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '7.5' to each side of the equation. -7.5 + 7.5 + b = -7.56306816 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + b = -7.56306816 + 7.5 b = -7.56306816 + 7.5 Combine like terms: -7.56306816 + 7.5 = -0.06306816 b = -0.06306816 Simplifying b = -0.06306816Solution
The solution to the problem is based on the solutions from the subproblems. b = {15.06306816, -0.06306816}
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