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Simplifying 10b2 + 9b + -13 = 10 Reorder the terms: -13 + 9b + 10b2 = 10 Solving -13 + 9b + 10b2 = 10 Solving for variable 'b'. Reorder the terms: -13 + -10 + 9b + 10b2 = 10 + -10 Combine like terms: -13 + -10 = -23 -23 + 9b + 10b2 = 10 + -10 Combine like terms: 10 + -10 = 0 -23 + 9b + 10b2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -2.3 + 0.9b + b2 = 0 Move the constant term to the right: Add '2.3' to each side of the equation. -2.3 + 0.9b + 2.3 + b2 = 0 + 2.3 Reorder the terms: -2.3 + 2.3 + 0.9b + b2 = 0 + 2.3 Combine like terms: -2.3 + 2.3 = 0.0 0.0 + 0.9b + b2 = 0 + 2.3 0.9b + b2 = 0 + 2.3 Combine like terms: 0 + 2.3 = 2.3 0.9b + b2 = 2.3 The b term is 0.9b. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9b + 0.2025 + b2 = 2.3 + 0.2025 Reorder the terms: 0.2025 + 0.9b + b2 = 2.3 + 0.2025 Combine like terms: 2.3 + 0.2025 = 2.5025 0.2025 + 0.9b + b2 = 2.5025 Factor a perfect square on the left side: (b + 0.45)(b + 0.45) = 2.5025 Calculate the square root of the right side: 1.581929202 Break this problem into two subproblems by setting (b + 0.45) equal to 1.581929202 and -1.581929202.Subproblem 1
b + 0.45 = 1.581929202 Simplifying b + 0.45 = 1.581929202 Reorder the terms: 0.45 + b = 1.581929202 Solving 0.45 + b = 1.581929202 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + b = 1.581929202 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + b = 1.581929202 + -0.45 b = 1.581929202 + -0.45 Combine like terms: 1.581929202 + -0.45 = 1.131929202 b = 1.131929202 Simplifying b = 1.131929202Subproblem 2
b + 0.45 = -1.581929202 Simplifying b + 0.45 = -1.581929202 Reorder the terms: 0.45 + b = -1.581929202 Solving 0.45 + b = -1.581929202 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + b = -1.581929202 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + b = -1.581929202 + -0.45 b = -1.581929202 + -0.45 Combine like terms: -1.581929202 + -0.45 = -2.031929202 b = -2.031929202 Simplifying b = -2.031929202Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.131929202, -2.031929202}
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