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Simplifying -5b(4 + -5b) = 19b + 4 (4 * -5b + -5b * -5b) = 19b + 4 (-20b + 25b2) = 19b + 4 Reorder the terms: -20b + 25b2 = 4 + 19b Solving -20b + 25b2 = 4 + 19b Solving for variable 'b'. Reorder the terms: -4 + -20b + -19b + 25b2 = 4 + 19b + -4 + -19b Combine like terms: -20b + -19b = -39b -4 + -39b + 25b2 = 4 + 19b + -4 + -19b Reorder the terms: -4 + -39b + 25b2 = 4 + -4 + 19b + -19b Combine like terms: 4 + -4 = 0 -4 + -39b + 25b2 = 0 + 19b + -19b -4 + -39b + 25b2 = 19b + -19b Combine like terms: 19b + -19b = 0 -4 + -39b + 25b2 = 0 Begin completing the square. Divide all terms by 25 the coefficient of the squared term: Divide each side by '25'. -0.16 + -1.56b + b2 = 0 Move the constant term to the right: Add '0.16' to each side of the equation. -0.16 + -1.56b + 0.16 + b2 = 0 + 0.16 Reorder the terms: -0.16 + 0.16 + -1.56b + b2 = 0 + 0.16 Combine like terms: -0.16 + 0.16 = 0.00 0.00 + -1.56b + b2 = 0 + 0.16 -1.56b + b2 = 0 + 0.16 Combine like terms: 0 + 0.16 = 0.16 -1.56b + b2 = 0.16 The b term is -1.56b. Take half its coefficient (-0.78). Square it (0.6084) and add it to both sides. Add '0.6084' to each side of the equation. -1.56b + 0.6084 + b2 = 0.16 + 0.6084 Reorder the terms: 0.6084 + -1.56b + b2 = 0.16 + 0.6084 Combine like terms: 0.16 + 0.6084 = 0.7684 0.6084 + -1.56b + b2 = 0.7684 Factor a perfect square on the left side: (b + -0.78)(b + -0.78) = 0.7684 Calculate the square root of the right side: 0.87658428 Break this problem into two subproblems by setting (b + -0.78) equal to 0.87658428 and -0.87658428.Subproblem 1
b + -0.78 = 0.87658428 Simplifying b + -0.78 = 0.87658428 Reorder the terms: -0.78 + b = 0.87658428 Solving -0.78 + b = 0.87658428 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.78' to each side of the equation. -0.78 + 0.78 + b = 0.87658428 + 0.78 Combine like terms: -0.78 + 0.78 = 0.00 0.00 + b = 0.87658428 + 0.78 b = 0.87658428 + 0.78 Combine like terms: 0.87658428 + 0.78 = 1.65658428 b = 1.65658428 Simplifying b = 1.65658428Subproblem 2
b + -0.78 = -0.87658428 Simplifying b + -0.78 = -0.87658428 Reorder the terms: -0.78 + b = -0.87658428 Solving -0.78 + b = -0.87658428 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.78' to each side of the equation. -0.78 + 0.78 + b = -0.87658428 + 0.78 Combine like terms: -0.78 + 0.78 = 0.00 0.00 + b = -0.87658428 + 0.78 b = -0.87658428 + 0.78 Combine like terms: -0.87658428 + 0.78 = -0.09658428 b = -0.09658428 Simplifying b = -0.09658428Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.65658428, -0.09658428}
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