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Simplifying 15b2 + -3b + -116 = 0 Reorder the terms: -116 + -3b + 15b2 = 0 Solving -116 + -3b + 15b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -7.733333333 + -0.2b + b2 = 0 Move the constant term to the right: Add '7.733333333' to each side of the equation. -7.733333333 + -0.2b + 7.733333333 + b2 = 0 + 7.733333333 Reorder the terms: -7.733333333 + 7.733333333 + -0.2b + b2 = 0 + 7.733333333 Combine like terms: -7.733333333 + 7.733333333 = 0.000000000 0.000000000 + -0.2b + b2 = 0 + 7.733333333 -0.2b + b2 = 0 + 7.733333333 Combine like terms: 0 + 7.733333333 = 7.733333333 -0.2b + b2 = 7.733333333 The b term is -0.2b. Take half its coefficient (-0.1). Square it (0.01) and add it to both sides. Add '0.01' to each side of the equation. -0.2b + 0.01 + b2 = 7.733333333 + 0.01 Reorder the terms: 0.01 + -0.2b + b2 = 7.733333333 + 0.01 Combine like terms: 7.733333333 + 0.01 = 7.743333333 0.01 + -0.2b + b2 = 7.743333333 Factor a perfect square on the left side: (b + -0.1)(b + -0.1) = 7.743333333 Calculate the square root of the right side: 2.782684555 Break this problem into two subproblems by setting (b + -0.1) equal to 2.782684555 and -2.782684555.Subproblem 1
b + -0.1 = 2.782684555 Simplifying b + -0.1 = 2.782684555 Reorder the terms: -0.1 + b = 2.782684555 Solving -0.1 + b = 2.782684555 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + b = 2.782684555 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + b = 2.782684555 + 0.1 b = 2.782684555 + 0.1 Combine like terms: 2.782684555 + 0.1 = 2.882684555 b = 2.882684555 Simplifying b = 2.882684555Subproblem 2
b + -0.1 = -2.782684555 Simplifying b + -0.1 = -2.782684555 Reorder the terms: -0.1 + b = -2.782684555 Solving -0.1 + b = -2.782684555 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.1' to each side of the equation. -0.1 + 0.1 + b = -2.782684555 + 0.1 Combine like terms: -0.1 + 0.1 = 0.0 0.0 + b = -2.782684555 + 0.1 b = -2.782684555 + 0.1 Combine like terms: -2.782684555 + 0.1 = -2.682684555 b = -2.682684555 Simplifying b = -2.682684555Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.882684555, -2.682684555}
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