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Simplifying (b + 5)(2b + 5) = 4b2 + b + 4 Reorder the terms: (5 + b)(2b + 5) = 4b2 + b + 4 Reorder the terms: (5 + b)(5 + 2b) = 4b2 + b + 4 Multiply (5 + b) * (5 + 2b) (5(5 + 2b) + b(5 + 2b)) = 4b2 + b + 4 ((5 * 5 + 2b * 5) + b(5 + 2b)) = 4b2 + b + 4 ((25 + 10b) + b(5 + 2b)) = 4b2 + b + 4 (25 + 10b + (5 * b + 2b * b)) = 4b2 + b + 4 (25 + 10b + (5b + 2b2)) = 4b2 + b + 4 Combine like terms: 10b + 5b = 15b (25 + 15b + 2b2) = 4b2 + b + 4 Reorder the terms: 25 + 15b + 2b2 = 4 + b + 4b2 Solving 25 + 15b + 2b2 = 4 + b + 4b2 Solving for variable 'b'. Reorder the terms: 25 + -4 + 15b + -1b + 2b2 + -4b2 = 4 + b + 4b2 + -4 + -1b + -4b2 Combine like terms: 25 + -4 = 21 21 + 15b + -1b + 2b2 + -4b2 = 4 + b + 4b2 + -4 + -1b + -4b2 Combine like terms: 15b + -1b = 14b 21 + 14b + 2b2 + -4b2 = 4 + b + 4b2 + -4 + -1b + -4b2 Combine like terms: 2b2 + -4b2 = -2b2 21 + 14b + -2b2 = 4 + b + 4b2 + -4 + -1b + -4b2 Reorder the terms: 21 + 14b + -2b2 = 4 + -4 + b + -1b + 4b2 + -4b2 Combine like terms: 4 + -4 = 0 21 + 14b + -2b2 = 0 + b + -1b + 4b2 + -4b2 21 + 14b + -2b2 = b + -1b + 4b2 + -4b2 Combine like terms: b + -1b = 0 21 + 14b + -2b2 = 0 + 4b2 + -4b2 21 + 14b + -2b2 = 4b2 + -4b2 Combine like terms: 4b2 + -4b2 = 0 21 + 14b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -10.5 + -7b + b2 = 0 Move the constant term to the right: Add '10.5' to each side of the equation. -10.5 + -7b + 10.5 + b2 = 0 + 10.5 Reorder the terms: -10.5 + 10.5 + -7b + b2 = 0 + 10.5 Combine like terms: -10.5 + 10.5 = 0.0 0.0 + -7b + b2 = 0 + 10.5 -7b + b2 = 0 + 10.5 Combine like terms: 0 + 10.5 = 10.5 -7b + b2 = 10.5 The b term is -7b. Take half its coefficient (-3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. -7b + 12.25 + b2 = 10.5 + 12.25 Reorder the terms: 12.25 + -7b + b2 = 10.5 + 12.25 Combine like terms: 10.5 + 12.25 = 22.75 12.25 + -7b + b2 = 22.75 Factor a perfect square on the left side: (b + -3.5)(b + -3.5) = 22.75 Calculate the square root of the right side: 4.769696007 Break this problem into two subproblems by setting (b + -3.5) equal to 4.769696007 and -4.769696007.Subproblem 1
b + -3.5 = 4.769696007 Simplifying b + -3.5 = 4.769696007 Reorder the terms: -3.5 + b = 4.769696007 Solving -3.5 + b = 4.769696007 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + b = 4.769696007 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + b = 4.769696007 + 3.5 b = 4.769696007 + 3.5 Combine like terms: 4.769696007 + 3.5 = 8.269696007 b = 8.269696007 Simplifying b = 8.269696007Subproblem 2
b + -3.5 = -4.769696007 Simplifying b + -3.5 = -4.769696007 Reorder the terms: -3.5 + b = -4.769696007 Solving -3.5 + b = -4.769696007 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + b = -4.769696007 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + b = -4.769696007 + 3.5 b = -4.769696007 + 3.5 Combine like terms: -4.769696007 + 3.5 = -1.269696007 b = -1.269696007 Simplifying b = -1.269696007Solution
The solution to the problem is based on the solutions from the subproblems. b = {8.269696007, -1.269696007}
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