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Simplifying 14b2 + -29b + -15 = 1 Reorder the terms: -15 + -29b + 14b2 = 1 Solving -15 + -29b + 14b2 = 1 Solving for variable 'b'. Reorder the terms: -15 + -1 + -29b + 14b2 = 1 + -1 Combine like terms: -15 + -1 = -16 -16 + -29b + 14b2 = 1 + -1 Combine like terms: 1 + -1 = 0 -16 + -29b + 14b2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -1.142857143 + -2.071428571b + b2 = 0 Move the constant term to the right: Add '1.142857143' to each side of the equation. -1.142857143 + -2.071428571b + 1.142857143 + b2 = 0 + 1.142857143 Reorder the terms: -1.142857143 + 1.142857143 + -2.071428571b + b2 = 0 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + -2.071428571b + b2 = 0 + 1.142857143 -2.071428571b + b2 = 0 + 1.142857143 Combine like terms: 0 + 1.142857143 = 1.142857143 -2.071428571b + b2 = 1.142857143 The b term is -2.071428571b. Take half its coefficient (-1.035714286). Square it (1.072704082) and add it to both sides. Add '1.072704082' to each side of the equation. -2.071428571b + 1.072704082 + b2 = 1.142857143 + 1.072704082 Reorder the terms: 1.072704082 + -2.071428571b + b2 = 1.142857143 + 1.072704082 Combine like terms: 1.142857143 + 1.072704082 = 2.215561225 1.072704082 + -2.071428571b + b2 = 2.215561225 Factor a perfect square on the left side: (b + -1.035714286)(b + -1.035714286) = 2.215561225 Calculate the square root of the right side: 1.488476142 Break this problem into two subproblems by setting (b + -1.035714286) equal to 1.488476142 and -1.488476142.Subproblem 1
b + -1.035714286 = 1.488476142 Simplifying b + -1.035714286 = 1.488476142 Reorder the terms: -1.035714286 + b = 1.488476142 Solving -1.035714286 + b = 1.488476142 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.035714286' to each side of the equation. -1.035714286 + 1.035714286 + b = 1.488476142 + 1.035714286 Combine like terms: -1.035714286 + 1.035714286 = 0.000000000 0.000000000 + b = 1.488476142 + 1.035714286 b = 1.488476142 + 1.035714286 Combine like terms: 1.488476142 + 1.035714286 = 2.524190428 b = 2.524190428 Simplifying b = 2.524190428Subproblem 2
b + -1.035714286 = -1.488476142 Simplifying b + -1.035714286 = -1.488476142 Reorder the terms: -1.035714286 + b = -1.488476142 Solving -1.035714286 + b = -1.488476142 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.035714286' to each side of the equation. -1.035714286 + 1.035714286 + b = -1.488476142 + 1.035714286 Combine like terms: -1.035714286 + 1.035714286 = 0.000000000 0.000000000 + b = -1.488476142 + 1.035714286 b = -1.488476142 + 1.035714286 Combine like terms: -1.488476142 + 1.035714286 = -0.452761856 b = -0.452761856 Simplifying b = -0.452761856Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.524190428, -0.452761856}
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