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Simplifying 4b2 + 8b + 7 = 14 Reorder the terms: 7 + 8b + 4b2 = 14 Solving 7 + 8b + 4b2 = 14 Solving for variable 'b'. Reorder the terms: 7 + -14 + 8b + 4b2 = 14 + -14 Combine like terms: 7 + -14 = -7 -7 + 8b + 4b2 = 14 + -14 Combine like terms: 14 + -14 = 0 -7 + 8b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + 2b + b2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + 2b + 1.75 + b2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + 2b + b2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + 2b + b2 = 0 + 1.75 2b + b2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 2b + b2 = 1.75 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 = 1.75 + 1 Reorder the terms: 1 + 2b + b2 = 1.75 + 1 Combine like terms: 1.75 + 1 = 2.75 1 + 2b + b2 = 2.75 Factor a perfect square on the left side: (b + 1)(b + 1) = 2.75 Calculate the square root of the right side: 1.658312395 Break this problem into two subproblems by setting (b + 1) equal to 1.658312395 and -1.658312395.Subproblem 1
b + 1 = 1.658312395 Simplifying b + 1 = 1.658312395 Reorder the terms: 1 + b = 1.658312395 Solving 1 + b = 1.658312395 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 = 1.658312395 + -1 Combine like terms: 1 + -1 = 0 0 + b = 1.658312395 + -1 b = 1.658312395 + -1 Combine like terms: 1.658312395 + -1 = 0.658312395 b = 0.658312395 Simplifying b = 0.658312395Subproblem 2
b + 1 = -1.658312395 Simplifying b + 1 = -1.658312395 Reorder the terms: 1 + b = -1.658312395 Solving 1 + b = -1.658312395 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 = -1.658312395 + -1 Combine like terms: 1 + -1 = 0 0 + b = -1.658312395 + -1 b = -1.658312395 + -1 Combine like terms: -1.658312395 + -1 = -2.658312395 b = -2.658312395 Simplifying b = -2.658312395Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.658312395, -2.658312395}
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