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Simplifying 7b2 + 32b + 6 = 0 Reorder the terms: 6 + 32b + 7b2 = 0 Solving 6 + 32b + 7b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.8571428571 + 4.571428571b + b2 = 0 Move the constant term to the right: Add '-0.8571428571' to each side of the equation. 0.8571428571 + 4.571428571b + -0.8571428571 + b2 = 0 + -0.8571428571 Reorder the terms: 0.8571428571 + -0.8571428571 + 4.571428571b + b2 = 0 + -0.8571428571 Combine like terms: 0.8571428571 + -0.8571428571 = 0.0000000000 0.0000000000 + 4.571428571b + b2 = 0 + -0.8571428571 4.571428571b + b2 = 0 + -0.8571428571 Combine like terms: 0 + -0.8571428571 = -0.8571428571 4.571428571b + b2 = -0.8571428571 The b term is 4.571428571b. Take half its coefficient (2.285714286). Square it (5.224489797) and add it to both sides. Add '5.224489797' to each side of the equation. 4.571428571b + 5.224489797 + b2 = -0.8571428571 + 5.224489797 Reorder the terms: 5.224489797 + 4.571428571b + b2 = -0.8571428571 + 5.224489797 Combine like terms: -0.8571428571 + 5.224489797 = 4.3673469399 5.224489797 + 4.571428571b + b2 = 4.3673469399 Factor a perfect square on the left side: (b + 2.285714286)(b + 2.285714286) = 4.3673469399 Calculate the square root of the right side: 2.089819834 Break this problem into two subproblems by setting (b + 2.285714286) equal to 2.089819834 and -2.089819834.Subproblem 1
b + 2.285714286 = 2.089819834 Simplifying b + 2.285714286 = 2.089819834 Reorder the terms: 2.285714286 + b = 2.089819834 Solving 2.285714286 + b = 2.089819834 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + b = 2.089819834 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + b = 2.089819834 + -2.285714286 b = 2.089819834 + -2.285714286 Combine like terms: 2.089819834 + -2.285714286 = -0.195894452 b = -0.195894452 Simplifying b = -0.195894452Subproblem 2
b + 2.285714286 = -2.089819834 Simplifying b + 2.285714286 = -2.089819834 Reorder the terms: 2.285714286 + b = -2.089819834 Solving 2.285714286 + b = -2.089819834 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.285714286' to each side of the equation. 2.285714286 + -2.285714286 + b = -2.089819834 + -2.285714286 Combine like terms: 2.285714286 + -2.285714286 = 0.000000000 0.000000000 + b = -2.089819834 + -2.285714286 b = -2.089819834 + -2.285714286 Combine like terms: -2.089819834 + -2.285714286 = -4.37553412 b = -4.37553412 Simplifying b = -4.37553412Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.195894452, -4.37553412}
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