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Simplifying 5b2 = -3b + 4b2 + 2 Reorder the terms: 5b2 = 2 + -3b + 4b2 Solving 5b2 = 2 + -3b + 4b2 Solving for variable 'b'. Reorder the terms: -2 + 3b + 5b2 + -4b2 = 2 + -3b + 4b2 + -2 + 3b + -4b2 Combine like terms: 5b2 + -4b2 = 1b2 -2 + 3b + 1b2 = 2 + -3b + 4b2 + -2 + 3b + -4b2 Reorder the terms: -2 + 3b + 1b2 = 2 + -2 + -3b + 3b + 4b2 + -4b2 Combine like terms: 2 + -2 = 0 -2 + 3b + 1b2 = 0 + -3b + 3b + 4b2 + -4b2 -2 + 3b + 1b2 = -3b + 3b + 4b2 + -4b2 Combine like terms: -3b + 3b = 0 -2 + 3b + 1b2 = 0 + 4b2 + -4b2 -2 + 3b + 1b2 = 4b2 + -4b2 Combine like terms: 4b2 + -4b2 = 0 -2 + 3b + 1b2 = 0 Begin completing the square. Move the constant term to the right: Add '2' to each side of the equation. -2 + 3b + 2 + b2 = 0 + 2 Reorder the terms: -2 + 2 + 3b + b2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 3b + b2 = 0 + 2 3b + b2 = 0 + 2 Combine like terms: 0 + 2 = 2 3b + b2 = 2 The b term is 3b. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3b + 2.25 + b2 = 2 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 2 + 2.25 Combine like terms: 2 + 2.25 = 4.25 2.25 + 3b + b2 = 4.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 4.25 Calculate the square root of the right side: 2.061552813 Break this problem into two subproblems by setting (b + 1.5) equal to 2.061552813 and -2.061552813.Subproblem 1
b + 1.5 = 2.061552813 Simplifying b + 1.5 = 2.061552813 Reorder the terms: 1.5 + b = 2.061552813 Solving 1.5 + b = 2.061552813 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 2.061552813 + -1.5 b = 2.061552813 + -1.5 Combine like terms: 2.061552813 + -1.5 = 0.561552813 b = 0.561552813 Simplifying b = 0.561552813Subproblem 2
b + 1.5 = -2.061552813 Simplifying b + 1.5 = -2.061552813 Reorder the terms: 1.5 + b = -2.061552813 Solving 1.5 + b = -2.061552813 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -2.061552813 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -2.061552813 + -1.5 b = -2.061552813 + -1.5 Combine like terms: -2.061552813 + -1.5 = -3.561552813 b = -3.561552813 Simplifying b = -3.561552813Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.561552813, -3.561552813}
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