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Simplifying 15b2 + 6b = 4b + 10b2 + 14 Reorder the terms: 6b + 15b2 = 4b + 10b2 + 14 Reorder the terms: 6b + 15b2 = 14 + 4b + 10b2 Solving 6b + 15b2 = 14 + 4b + 10b2 Solving for variable 'b'. Reorder the terms: -14 + 6b + -4b + 15b2 + -10b2 = 14 + 4b + 10b2 + -14 + -4b + -10b2 Combine like terms: 6b + -4b = 2b -14 + 2b + 15b2 + -10b2 = 14 + 4b + 10b2 + -14 + -4b + -10b2 Combine like terms: 15b2 + -10b2 = 5b2 -14 + 2b + 5b2 = 14 + 4b + 10b2 + -14 + -4b + -10b2 Reorder the terms: -14 + 2b + 5b2 = 14 + -14 + 4b + -4b + 10b2 + -10b2 Combine like terms: 14 + -14 = 0 -14 + 2b + 5b2 = 0 + 4b + -4b + 10b2 + -10b2 -14 + 2b + 5b2 = 4b + -4b + 10b2 + -10b2 Combine like terms: 4b + -4b = 0 -14 + 2b + 5b2 = 0 + 10b2 + -10b2 -14 + 2b + 5b2 = 10b2 + -10b2 Combine like terms: 10b2 + -10b2 = 0 -14 + 2b + 5b2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -2.8 + 0.4b + b2 = 0 Move the constant term to the right: Add '2.8' to each side of the equation. -2.8 + 0.4b + 2.8 + b2 = 0 + 2.8 Reorder the terms: -2.8 + 2.8 + 0.4b + b2 = 0 + 2.8 Combine like terms: -2.8 + 2.8 = 0.0 0.0 + 0.4b + b2 = 0 + 2.8 0.4b + b2 = 0 + 2.8 Combine like terms: 0 + 2.8 = 2.8 0.4b + b2 = 2.8 The b term is 0.4b. Take half its coefficient (0.2). Square it (0.04) and add it to both sides. Add '0.04' to each side of the equation. 0.4b + 0.04 + b2 = 2.8 + 0.04 Reorder the terms: 0.04 + 0.4b + b2 = 2.8 + 0.04 Combine like terms: 2.8 + 0.04 = 2.84 0.04 + 0.4b + b2 = 2.84 Factor a perfect square on the left side: (b + 0.2)(b + 0.2) = 2.84 Calculate the square root of the right side: 1.685229955 Break this problem into two subproblems by setting (b + 0.2) equal to 1.685229955 and -1.685229955.Subproblem 1
b + 0.2 = 1.685229955 Simplifying b + 0.2 = 1.685229955 Reorder the terms: 0.2 + b = 1.685229955 Solving 0.2 + b = 1.685229955 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + b = 1.685229955 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + b = 1.685229955 + -0.2 b = 1.685229955 + -0.2 Combine like terms: 1.685229955 + -0.2 = 1.485229955 b = 1.485229955 Simplifying b = 1.485229955Subproblem 2
b + 0.2 = -1.685229955 Simplifying b + 0.2 = -1.685229955 Reorder the terms: 0.2 + b = -1.685229955 Solving 0.2 + b = -1.685229955 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2' to each side of the equation. 0.2 + -0.2 + b = -1.685229955 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + b = -1.685229955 + -0.2 b = -1.685229955 + -0.2 Combine like terms: -1.685229955 + -0.2 = -1.885229955 b = -1.885229955 Simplifying b = -1.885229955Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.485229955, -1.885229955}
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