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Simplifying 4b2 + 8b + 7 = 42 Reorder the terms: 7 + 8b + 4b2 = 42 Solving 7 + 8b + 4b2 = 42 Solving for variable 'b'. Reorder the terms: 7 + -42 + 8b + 4b2 = 42 + -42 Combine like terms: 7 + -42 = -35 -35 + 8b + 4b2 = 42 + -42 Combine like terms: 42 + -42 = 0 -35 + 8b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -8.75 + 2b + b2 = 0 Move the constant term to the right: Add '8.75' to each side of the equation. -8.75 + 2b + 8.75 + b2 = 0 + 8.75 Reorder the terms: -8.75 + 8.75 + 2b + b2 = 0 + 8.75 Combine like terms: -8.75 + 8.75 = 0.00 0.00 + 2b + b2 = 0 + 8.75 2b + b2 = 0 + 8.75 Combine like terms: 0 + 8.75 = 8.75 2b + b2 = 8.75 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 8.75 + 1 Reorder the terms: 1 + 2b + b2 = 8.75 + 1 Combine like terms: 8.75 + 1 = 9.75 1 + 2b + b2 = 9.75 Factor a perfect square on the left side: (b + 1)(b + 1) = 9.75 Calculate the square root of the right side: 3.122498999 Break this problem into two subproblems by setting (b + 1) equal to 3.122498999 and -3.122498999.Subproblem 1
b + 1 = 3.122498999 Simplifying b + 1 = 3.122498999 Reorder the terms: 1 + b = 3.122498999 Solving 1 + b = 3.122498999 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 3.122498999 + -1 Combine like terms: 1 + -1 = 0 0 + b = 3.122498999 + -1 b = 3.122498999 + -1 Combine like terms: 3.122498999 + -1 = 2.122498999 b = 2.122498999 Simplifying b = 2.122498999Subproblem 2
b + 1 = -3.122498999 Simplifying b + 1 = -3.122498999 Reorder the terms: 1 + b = -3.122498999 Solving 1 + b = -3.122498999 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -3.122498999 + -1 Combine like terms: 1 + -1 = 0 0 + b = -3.122498999 + -1 b = -3.122498999 + -1 Combine like terms: -3.122498999 + -1 = -4.122498999 b = -4.122498999 Simplifying b = -4.122498999Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.122498999, -4.122498999}
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