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Simplifying 100b2 + -258b + 144 = 5b2 Reorder the terms: 144 + -258b + 100b2 = 5b2 Solving 144 + -258b + 100b2 = 5b2 Solving for variable 'b'. Combine like terms: 100b2 + -5b2 = 95b2 144 + -258b + 95b2 = 5b2 + -5b2 Combine like terms: 5b2 + -5b2 = 0 144 + -258b + 95b2 = 0 Begin completing the square. Divide all terms by 95 the coefficient of the squared term: Divide each side by '95'. 1.515789474 + -2.715789474b + b2 = 0 Move the constant term to the right: Add '-1.515789474' to each side of the equation. 1.515789474 + -2.715789474b + -1.515789474 + b2 = 0 + -1.515789474 Reorder the terms: 1.515789474 + -1.515789474 + -2.715789474b + b2 = 0 + -1.515789474 Combine like terms: 1.515789474 + -1.515789474 = 0.000000000 0.000000000 + -2.715789474b + b2 = 0 + -1.515789474 -2.715789474b + b2 = 0 + -1.515789474 Combine like terms: 0 + -1.515789474 = -1.515789474 -2.715789474b + b2 = -1.515789474 The b term is -2.715789474b. Take half its coefficient (-1.357894737). Square it (1.843878117) and add it to both sides. Add '1.843878117' to each side of the equation. -2.715789474b + 1.843878117 + b2 = -1.515789474 + 1.843878117 Reorder the terms: 1.843878117 + -2.715789474b + b2 = -1.515789474 + 1.843878117 Combine like terms: -1.515789474 + 1.843878117 = 0.328088643 1.843878117 + -2.715789474b + b2 = 0.328088643 Factor a perfect square on the left side: (b + -1.357894737)(b + -1.357894737) = 0.328088643 Calculate the square root of the right side: 0.572790226 Break this problem into two subproblems by setting (b + -1.357894737) equal to 0.572790226 and -0.572790226.Subproblem 1
b + -1.357894737 = 0.572790226 Simplifying b + -1.357894737 = 0.572790226 Reorder the terms: -1.357894737 + b = 0.572790226 Solving -1.357894737 + b = 0.572790226 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.357894737' to each side of the equation. -1.357894737 + 1.357894737 + b = 0.572790226 + 1.357894737 Combine like terms: -1.357894737 + 1.357894737 = 0.000000000 0.000000000 + b = 0.572790226 + 1.357894737 b = 0.572790226 + 1.357894737 Combine like terms: 0.572790226 + 1.357894737 = 1.930684963 b = 1.930684963 Simplifying b = 1.930684963Subproblem 2
b + -1.357894737 = -0.572790226 Simplifying b + -1.357894737 = -0.572790226 Reorder the terms: -1.357894737 + b = -0.572790226 Solving -1.357894737 + b = -0.572790226 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.357894737' to each side of the equation. -1.357894737 + 1.357894737 + b = -0.572790226 + 1.357894737 Combine like terms: -1.357894737 + 1.357894737 = 0.000000000 0.000000000 + b = -0.572790226 + 1.357894737 b = -0.572790226 + 1.357894737 Combine like terms: -0.572790226 + 1.357894737 = 0.785104511 b = 0.785104511 Simplifying b = 0.785104511Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.930684963, 0.785104511}
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