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Simplifying 6(2b + -4) * b = 5 Reorder the terms: 6(-4 + 2b) * b = 5 Reorder the terms for easier multiplication: 6b(-4 + 2b) = 5 (-4 * 6b + 2b * 6b) = 5 (-24b + 12b2) = 5 Solving -24b + 12b2 = 5 Solving for variable 'b'. Reorder the terms: -5 + -24b + 12b2 = 5 + -5 Combine like terms: 5 + -5 = 0 -5 + -24b + 12b2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.4166666667 + -2b + b2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + -2b + 0.4166666667 + b2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + -2b + b2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + -2b + b2 = 0 + 0.4166666667 -2b + b2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 -2b + b2 = 0.4166666667 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 = 0.4166666667 + 1 Reorder the terms: 1 + -2b + b2 = 0.4166666667 + 1 Combine like terms: 0.4166666667 + 1 = 1.4166666667 1 + -2b + b2 = 1.4166666667 Factor a perfect square on the left side: (b + -1)(b + -1) = 1.4166666667 Calculate the square root of the right side: 1.190238071 Break this problem into two subproblems by setting (b + -1) equal to 1.190238071 and -1.190238071.Subproblem 1
b + -1 = 1.190238071 Simplifying b + -1 = 1.190238071 Reorder the terms: -1 + b = 1.190238071 Solving -1 + b = 1.190238071 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 = 1.190238071 + 1 Combine like terms: -1 + 1 = 0 0 + b = 1.190238071 + 1 b = 1.190238071 + 1 Combine like terms: 1.190238071 + 1 = 2.190238071 b = 2.190238071 Simplifying b = 2.190238071Subproblem 2
b + -1 = -1.190238071 Simplifying b + -1 = -1.190238071 Reorder the terms: -1 + b = -1.190238071 Solving -1 + b = -1.190238071 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 = -1.190238071 + 1 Combine like terms: -1 + 1 = 0 0 + b = -1.190238071 + 1 b = -1.190238071 + 1 Combine like terms: -1.190238071 + 1 = -0.190238071 b = -0.190238071 Simplifying b = -0.190238071Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.190238071, -0.190238071}
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