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Simplifying -7(b2 + 2b + -5) + b(2b + 6) + 8 = 0 Reorder the terms: -7(-5 + 2b + b2) + b(2b + 6) + 8 = 0 (-5 * -7 + 2b * -7 + b2 * -7) + b(2b + 6) + 8 = 0 (35 + -14b + -7b2) + b(2b + 6) + 8 = 0 Reorder the terms: 35 + -14b + -7b2 + b(6 + 2b) + 8 = 0 35 + -14b + -7b2 + (6 * b + 2b * b) + 8 = 0 35 + -14b + -7b2 + (6b + 2b2) + 8 = 0 Reorder the terms: 35 + 8 + -14b + 6b + -7b2 + 2b2 = 0 Combine like terms: 35 + 8 = 43 43 + -14b + 6b + -7b2 + 2b2 = 0 Combine like terms: -14b + 6b = -8b 43 + -8b + -7b2 + 2b2 = 0 Combine like terms: -7b2 + 2b2 = -5b2 43 + -8b + -5b2 = 0 Solving 43 + -8b + -5b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -8.6 + 1.6b + b2 = 0 Move the constant term to the right: Add '8.6' to each side of the equation. -8.6 + 1.6b + 8.6 + b2 = 0 + 8.6 Reorder the terms: -8.6 + 8.6 + 1.6b + b2 = 0 + 8.6 Combine like terms: -8.6 + 8.6 = 0.0 0.0 + 1.6b + b2 = 0 + 8.6 1.6b + b2 = 0 + 8.6 Combine like terms: 0 + 8.6 = 8.6 1.6b + b2 = 8.6 The b term is 1.6b. Take half its coefficient (0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. 1.6b + 0.64 + b2 = 8.6 + 0.64 Reorder the terms: 0.64 + 1.6b + b2 = 8.6 + 0.64 Combine like terms: 8.6 + 0.64 = 9.24 0.64 + 1.6b + b2 = 9.24 Factor a perfect square on the left side: (b + 0.8)(b + 0.8) = 9.24 Calculate the square root of the right side: 3.039736831 Break this problem into two subproblems by setting (b + 0.8) equal to 3.039736831 and -3.039736831.Subproblem 1
b + 0.8 = 3.039736831 Simplifying b + 0.8 = 3.039736831 Reorder the terms: 0.8 + b = 3.039736831 Solving 0.8 + b = 3.039736831 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + b = 3.039736831 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + b = 3.039736831 + -0.8 b = 3.039736831 + -0.8 Combine like terms: 3.039736831 + -0.8 = 2.239736831 b = 2.239736831 Simplifying b = 2.239736831Subproblem 2
b + 0.8 = -3.039736831 Simplifying b + 0.8 = -3.039736831 Reorder the terms: 0.8 + b = -3.039736831 Solving 0.8 + b = -3.039736831 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.8' to each side of the equation. 0.8 + -0.8 + b = -3.039736831 + -0.8 Combine like terms: 0.8 + -0.8 = 0.0 0.0 + b = -3.039736831 + -0.8 b = -3.039736831 + -0.8 Combine like terms: -3.039736831 + -0.8 = -3.839736831 b = -3.839736831 Simplifying b = -3.839736831Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.239736831, -3.839736831}
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