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Simplifying 12b2 + 25b + 1 = 0 Reorder the terms: 1 + 25b + 12b2 = 0 Solving 1 + 25b + 12b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.08333333333 + 2.083333333b + b2 = 0 Move the constant term to the right: Add '-0.08333333333' to each side of the equation. 0.08333333333 + 2.083333333b + -0.08333333333 + b2 = 0 + -0.08333333333 Reorder the terms: 0.08333333333 + -0.08333333333 + 2.083333333b + b2 = 0 + -0.08333333333 Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000 0.00000000000 + 2.083333333b + b2 = 0 + -0.08333333333 2.083333333b + b2 = 0 + -0.08333333333 Combine like terms: 0 + -0.08333333333 = -0.08333333333 2.083333333b + b2 = -0.08333333333 The b term is 2.083333333b. Take half its coefficient (1.041666667). Square it (1.085069445) and add it to both sides. Add '1.085069445' to each side of the equation. 2.083333333b + 1.085069445 + b2 = -0.08333333333 + 1.085069445 Reorder the terms: 1.085069445 + 2.083333333b + b2 = -0.08333333333 + 1.085069445 Combine like terms: -0.08333333333 + 1.085069445 = 1.00173611167 1.085069445 + 2.083333333b + b2 = 1.00173611167 Factor a perfect square on the left side: (b + 1.041666667)(b + 1.041666667) = 1.00173611167 Calculate the square root of the right side: 1.000867679 Break this problem into two subproblems by setting (b + 1.041666667) equal to 1.000867679 and -1.000867679.Subproblem 1
b + 1.041666667 = 1.000867679 Simplifying b + 1.041666667 = 1.000867679 Reorder the terms: 1.041666667 + b = 1.000867679 Solving 1.041666667 + b = 1.000867679 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.041666667' to each side of the equation. 1.041666667 + -1.041666667 + b = 1.000867679 + -1.041666667 Combine like terms: 1.041666667 + -1.041666667 = 0.000000000 0.000000000 + b = 1.000867679 + -1.041666667 b = 1.000867679 + -1.041666667 Combine like terms: 1.000867679 + -1.041666667 = -0.040798988 b = -0.040798988 Simplifying b = -0.040798988Subproblem 2
b + 1.041666667 = -1.000867679 Simplifying b + 1.041666667 = -1.000867679 Reorder the terms: 1.041666667 + b = -1.000867679 Solving 1.041666667 + b = -1.000867679 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.041666667' to each side of the equation. 1.041666667 + -1.041666667 + b = -1.000867679 + -1.041666667 Combine like terms: 1.041666667 + -1.041666667 = 0.000000000 0.000000000 + b = -1.000867679 + -1.041666667 b = -1.000867679 + -1.041666667 Combine like terms: -1.000867679 + -1.041666667 = -2.042534346 b = -2.042534346 Simplifying b = -2.042534346Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.040798988, -2.042534346}
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