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Simplifying b = 1.5(b + 2) * 15(2b + -9) Reorder the terms: b = 1.5(2 + b) * 15(2b + -9) Reorder the terms: b = 1.5(2 + b) * 15(-9 + 2b) Reorder the terms for easier multiplication: b = 1.5 * 15(2 + b)(-9 + 2b) Multiply 1.5 * 15 b = 22.5(2 + b)(-9 + 2b) Multiply (2 + b) * (-9 + 2b) b = 22.5(2(-9 + 2b) + b(-9 + 2b)) b = 22.5((-9 * 2 + 2b * 2) + b(-9 + 2b)) b = 22.5((-18 + 4b) + b(-9 + 2b)) b = 22.5(-18 + 4b + (-9 * b + 2b * b)) b = 22.5(-18 + 4b + (-9b + 2b2)) Combine like terms: 4b + -9b = -5b b = 22.5(-18 + -5b + 2b2) b = (-18 * 22.5 + -5b * 22.5 + 2b2 * 22.5) b = (-405 + -112.5b + 45b2) Solving b = -405 + -112.5b + 45b2 Solving for variable 'b'. Reorder the terms: 405 + b + 112.5b + -45b2 = -405 + -112.5b + 45b2 + 405 + 112.5b + -45b2 Combine like terms: b + 112.5b = 113.5b 405 + 113.5b + -45b2 = -405 + -112.5b + 45b2 + 405 + 112.5b + -45b2 Reorder the terms: 405 + 113.5b + -45b2 = -405 + 405 + -112.5b + 112.5b + 45b2 + -45b2 Combine like terms: -405 + 405 = 0 405 + 113.5b + -45b2 = 0 + -112.5b + 112.5b + 45b2 + -45b2 405 + 113.5b + -45b2 = -112.5b + 112.5b + 45b2 + -45b2 Combine like terms: -112.5b + 112.5b = 0.0 405 + 113.5b + -45b2 = 0.0 + 45b2 + -45b2 405 + 113.5b + -45b2 = 45b2 + -45b2 Combine like terms: 45b2 + -45b2 = 0 405 + 113.5b + -45b2 = 0 Begin completing the square. Divide all terms by -45 the coefficient of the squared term: Divide each side by '-45'. -9 + -2.522222222b + b2 = 0 Move the constant term to the right: Add '9' to each side of the equation. -9 + -2.522222222b + 9 + b2 = 0 + 9 Reorder the terms: -9 + 9 + -2.522222222b + b2 = 0 + 9 Combine like terms: -9 + 9 = 0 0 + -2.522222222b + b2 = 0 + 9 -2.522222222b + b2 = 0 + 9 Combine like terms: 0 + 9 = 9 -2.522222222b + b2 = 9 The b term is -2.522222222b. Take half its coefficient (-1.261111111). Square it (1.590401234) and add it to both sides. Add '1.590401234' to each side of the equation. -2.522222222b + 1.590401234 + b2 = 9 + 1.590401234 Reorder the terms: 1.590401234 + -2.522222222b + b2 = 9 + 1.590401234 Combine like terms: 9 + 1.590401234 = 10.590401234 1.590401234 + -2.522222222b + b2 = 10.590401234 Factor a perfect square on the left side: (b + -1.261111111)(b + -1.261111111) = 10.590401234 Calculate the square root of the right side: 3.254289667 Break this problem into two subproblems by setting (b + -1.261111111) equal to 3.254289667 and -3.254289667.Subproblem 1
b + -1.261111111 = 3.254289667 Simplifying b + -1.261111111 = 3.254289667 Reorder the terms: -1.261111111 + b = 3.254289667 Solving -1.261111111 + b = 3.254289667 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.261111111' to each side of the equation. -1.261111111 + 1.261111111 + b = 3.254289667 + 1.261111111 Combine like terms: -1.261111111 + 1.261111111 = 0.000000000 0.000000000 + b = 3.254289667 + 1.261111111 b = 3.254289667 + 1.261111111 Combine like terms: 3.254289667 + 1.261111111 = 4.515400778 b = 4.515400778 Simplifying b = 4.515400778Subproblem 2
b + -1.261111111 = -3.254289667 Simplifying b + -1.261111111 = -3.254289667 Reorder the terms: -1.261111111 + b = -3.254289667 Solving -1.261111111 + b = -3.254289667 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.261111111' to each side of the equation. -1.261111111 + 1.261111111 + b = -3.254289667 + 1.261111111 Combine like terms: -1.261111111 + 1.261111111 = 0.000000000 0.000000000 + b = -3.254289667 + 1.261111111 b = -3.254289667 + 1.261111111 Combine like terms: -3.254289667 + 1.261111111 = -1.993178556 b = -1.993178556 Simplifying b = -1.993178556Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.515400778, -1.993178556}
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