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Simplifying 10b2 + 3b = 75 Reorder the terms: 3b + 10b2 = 75 Solving 3b + 10b2 = 75 Solving for variable 'b'. Reorder the terms: -75 + 3b + 10b2 = 75 + -75 Combine like terms: 75 + -75 = 0 -75 + 3b + 10b2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -7.5 + 0.3b + b2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 0.3b + 7.5 + b2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 0.3b + b2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 0.3b + b2 = 0 + 7.5 0.3b + b2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 0.3b + b2 = 7.5 The b term is 0.3b. Take half its coefficient (0.15). Square it (0.0225) and add it to both sides. Add '0.0225' to each side of the equation. 0.3b + 0.0225 + b2 = 7.5 + 0.0225 Reorder the terms: 0.0225 + 0.3b + b2 = 7.5 + 0.0225 Combine like terms: 7.5 + 0.0225 = 7.5225 0.0225 + 0.3b + b2 = 7.5225 Factor a perfect square on the left side: (b + 0.15)(b + 0.15) = 7.5225 Calculate the square root of the right side: 2.74271763 Break this problem into two subproblems by setting (b + 0.15) equal to 2.74271763 and -2.74271763.Subproblem 1
b + 0.15 = 2.74271763 Simplifying b + 0.15 = 2.74271763 Reorder the terms: 0.15 + b = 2.74271763 Solving 0.15 + b = 2.74271763 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + b = 2.74271763 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + b = 2.74271763 + -0.15 b = 2.74271763 + -0.15 Combine like terms: 2.74271763 + -0.15 = 2.59271763 b = 2.59271763 Simplifying b = 2.59271763Subproblem 2
b + 0.15 = -2.74271763 Simplifying b + 0.15 = -2.74271763 Reorder the terms: 0.15 + b = -2.74271763 Solving 0.15 + b = -2.74271763 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.15' to each side of the equation. 0.15 + -0.15 + b = -2.74271763 + -0.15 Combine like terms: 0.15 + -0.15 = 0.00 0.00 + b = -2.74271763 + -0.15 b = -2.74271763 + -0.15 Combine like terms: -2.74271763 + -0.15 = -2.89271763 b = -2.89271763 Simplifying b = -2.89271763Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.59271763, -2.89271763}
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