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Simplifying 5(3b2 + -2b) + 10b2 + -1 = 0 Reorder the terms: 5(-2b + 3b2) + 10b2 + -1 = 0 (-2b * 5 + 3b2 * 5) + 10b2 + -1 = 0 (-10b + 15b2) + 10b2 + -1 = 0 Reorder the terms: -1 + -10b + 15b2 + 10b2 = 0 Combine like terms: 15b2 + 10b2 = 25b2 -1 + -10b + 25b2 = 0 Solving -1 + -10b + 25b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.04 + -0.4b + b2 = 0 Move the constant term to the right: Add '0.04' to each side of the equation. -0.04 + -0.4b + 0.04 + b2 = 0 + 0.04 Reorder the terms: -0.04 + 0.04 + -0.4b + b2 = 0 + 0.04 Combine like terms: -0.04 + 0.04 = 0.00 0.00 + -0.4b + b2 = 0 + 0.04 -0.4b + b2 = 0 + 0.04 Combine like terms: 0 + 0.04 = 0.04 -0.4b + b2 = 0.04 The b term is -0.4b. Take half its coefficient (-0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. -0.4b + 0.04 + b2 = 0.04 + 0.04 Reorder the terms: 0.04 + -0.4b + b2 = 0.04 + 0.04 Combine like terms: 0.04 + 0.04 = 0.08 0.04 + -0.4b + b2 = 0.08 Factor a perfect square on the left side: (b + -0.2)(b + -0.2) = 0.08 Calculate the square root of the right side: 0.282842712 Break this problem into two subproblems by setting (b + -0.2) equal to 0.282842712 and -0.282842712.Subproblem 1
b + -0.2 = 0.282842712 Simplifying b + -0.2 = 0.282842712 Reorder the terms: -0.2 + b = 0.282842712 Solving -0.2 + b = 0.282842712 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + b = 0.282842712 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + b = 0.282842712 + 0.2 b = 0.282842712 + 0.2 Combine like terms: 0.282842712 + 0.2 = 0.482842712 b = 0.482842712 Simplifying b = 0.482842712Subproblem 2
b + -0.2 = -0.282842712 Simplifying b + -0.2 = -0.282842712 Reorder the terms: -0.2 + b = -0.282842712 Solving -0.2 + b = -0.282842712 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.2' to each side of the equation. -0.2 + 0.2 + b = -0.282842712 + 0.2 Combine like terms: -0.2 + 0.2 = 0.0 0.0 + b = -0.282842712 + 0.2 b = -0.282842712 + 0.2 Combine like terms: -0.282842712 + 0.2 = -0.082842712 b = -0.082842712 Simplifying b = -0.082842712Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.482842712, -0.082842712}
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