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Simplifying 10 = b(2b + -3) Reorder the terms: 10 = b(-3 + 2b) 10 = (-3 * b + 2b * b) 10 = (-3b + 2b2) Solving 10 = -3b + 2b2 Solving for variable 'b'. Reorder the terms: 10 + 3b + -2b2 = -3b + 3b + 2b2 + -2b2 Combine like terms: -3b + 3b = 0 10 + 3b + -2b2 = 0 + 2b2 + -2b2 10 + 3b + -2b2 = 2b2 + -2b2 Combine like terms: 2b2 + -2b2 = 0 10 + 3b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -5 + -1.5b + b2 = 0 Move the constant term to the right: Add '5' to each side of the equation. -5 + -1.5b + 5 + b2 = 0 + 5 Reorder the terms: -5 + 5 + -1.5b + b2 = 0 + 5 Combine like terms: -5 + 5 = 0 0 + -1.5b + b2 = 0 + 5 -1.5b + b2 = 0 + 5 Combine like terms: 0 + 5 = 5 -1.5b + b2 = 5 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 = 5 + 0.5625 Reorder the terms: 0.5625 + -1.5b + b2 = 5 + 0.5625 Combine like terms: 5 + 0.5625 = 5.5625 0.5625 + -1.5b + b2 = 5.5625 Factor a perfect square on the left side: (b + -0.75)(b + -0.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (b + -0.75) equal to 2.358495283 and -2.358495283.Subproblem 1
b + -0.75 = 2.358495283 Simplifying b + -0.75 = 2.358495283 Reorder the terms: -0.75 + b = 2.358495283 Solving -0.75 + b = 2.358495283 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 = 2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = 2.358495283 + 0.75 b = 2.358495283 + 0.75 Combine like terms: 2.358495283 + 0.75 = 3.108495283 b = 3.108495283 Simplifying b = 3.108495283Subproblem 2
b + -0.75 = -2.358495283 Simplifying b + -0.75 = -2.358495283 Reorder the terms: -0.75 + b = -2.358495283 Solving -0.75 + b = -2.358495283 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 = -2.358495283 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = -2.358495283 + 0.75 b = -2.358495283 + 0.75 Combine like terms: -2.358495283 + 0.75 = -1.608495283 b = -1.608495283 Simplifying b = -1.608495283Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.108495283, -1.608495283}
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