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Simplifying 10b2 + -15b = 8 + -12 Reorder the terms: -15b + 10b2 = 8 + -12 Combine like terms: 8 + -12 = -4 -15b + 10b2 = -4 Solving -15b + 10b2 = -4 Solving for variable 'b'. Reorder the terms: 4 + -15b + 10b2 = -4 + 4 Combine like terms: -4 + 4 = 0 4 + -15b + 10b2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.4 + -1.5b + b2 = 0 Move the constant term to the right: Add '-0.4' to each side of the equation. 0.4 + -1.5b + -0.4 + b2 = 0 + -0.4 Reorder the terms: 0.4 + -0.4 + -1.5b + b2 = 0 + -0.4 Combine like terms: 0.4 + -0.4 = 0.0 0.0 + -1.5b + b2 = 0 + -0.4 -1.5b + b2 = 0 + -0.4 Combine like terms: 0 + -0.4 = -0.4 -1.5b + b2 = -0.4 The b term is -1.5b. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5b + 0.5625 + b2 = -0.4 + 0.5625 Reorder the terms: 0.5625 + -1.5b + b2 = -0.4 + 0.5625 Combine like terms: -0.4 + 0.5625 = 0.1625 0.5625 + -1.5b + b2 = 0.1625 Factor a perfect square on the left side: (b + -0.75)(b + -0.75) = 0.1625 Calculate the square root of the right side: 0.403112887 Break this problem into two subproblems by setting (b + -0.75) equal to 0.403112887 and -0.403112887.Subproblem 1
b + -0.75 = 0.403112887 Simplifying b + -0.75 = 0.403112887 Reorder the terms: -0.75 + b = 0.403112887 Solving -0.75 + b = 0.403112887 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = 0.403112887 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = 0.403112887 + 0.75 b = 0.403112887 + 0.75 Combine like terms: 0.403112887 + 0.75 = 1.153112887 b = 1.153112887 Simplifying b = 1.153112887Subproblem 2
b + -0.75 = -0.403112887 Simplifying b + -0.75 = -0.403112887 Reorder the terms: -0.75 + b = -0.403112887 Solving -0.75 + b = -0.403112887 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = -0.403112887 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = -0.403112887 + 0.75 b = -0.403112887 + 0.75 Combine like terms: -0.403112887 + 0.75 = 0.346887113 b = 0.346887113 Simplifying b = 0.346887113Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.153112887, 0.346887113}
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