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Simplifying 150b = -35b2 + 25 Reorder the terms: 150b = 25 + -35b2 Solving 150b = 25 + -35b2 Solving for variable 'b'. Reorder the terms: -25 + 150b + 35b2 = 25 + -35b2 + -25 + 35b2 Reorder the terms: -25 + 150b + 35b2 = 25 + -25 + -35b2 + 35b2 Combine like terms: 25 + -25 = 0 -25 + 150b + 35b2 = 0 + -35b2 + 35b2 -25 + 150b + 35b2 = -35b2 + 35b2 Combine like terms: -35b2 + 35b2 = 0 -25 + 150b + 35b2 = 0 Factor out the Greatest Common Factor (GCF), '5'. 5(-5 + 30b + 7b2) = 0 Ignore the factor 5.Subproblem 1
Set the factor '(-5 + 30b + 7b2)' equal to zero and attempt to solve: Simplifying -5 + 30b + 7b2 = 0 Solving -5 + 30b + 7b2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.7142857143 + 4.285714286b + b2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 4.285714286b + 0.7142857143 + b2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 4.285714286b + b2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 4.285714286b + b2 = 0 + 0.7142857143 4.285714286b + b2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 4.285714286b + b2 = 0.7142857143 The b term is 4.285714286b. Take half its coefficient (2.142857143). Square it (4.591836735) and add it to both sides. Add '4.591836735' to each side of the equation. 4.285714286b + 4.591836735 + b2 = 0.7142857143 + 4.591836735 Reorder the terms: 4.591836735 + 4.285714286b + b2 = 0.7142857143 + 4.591836735 Combine like terms: 0.7142857143 + 4.591836735 = 5.3061224493 4.591836735 + 4.285714286b + b2 = 5.3061224493 Factor a perfect square on the left side: (b + 2.142857143)(b + 2.142857143) = 5.3061224493 Calculate the square root of the right side: 2.303502214 Break this problem into two subproblems by setting (b + 2.142857143) equal to 2.303502214 and -2.303502214.Subproblem 1
b + 2.142857143 = 2.303502214 Simplifying b + 2.142857143 = 2.303502214 Reorder the terms: 2.142857143 + b = 2.303502214 Solving 2.142857143 + b = 2.303502214 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + b = 2.303502214 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + b = 2.303502214 + -2.142857143 b = 2.303502214 + -2.142857143 Combine like terms: 2.303502214 + -2.142857143 = 0.160645071 b = 0.160645071 Simplifying b = 0.160645071Subproblem 2
b + 2.142857143 = -2.303502214 Simplifying b + 2.142857143 = -2.303502214 Reorder the terms: 2.142857143 + b = -2.303502214 Solving 2.142857143 + b = -2.303502214 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.142857143' to each side of the equation. 2.142857143 + -2.142857143 + b = -2.303502214 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + b = -2.303502214 + -2.142857143 b = -2.303502214 + -2.142857143 Combine like terms: -2.303502214 + -2.142857143 = -4.446359357 b = -4.446359357 Simplifying b = -4.446359357Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.160645071, -4.446359357}Solution
b = {0.160645071, -4.446359357}
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