64 x 2 - 16 a + 4 b x + a b = 0 In the given equation, a

Question

64 x 2 - 16 a + 4 b x + a b = 0 In the given equation, a and b are positive constants. The sum of the solutions to the given equation is &#160; k 4 a + b , where k is a constant. What is the value of k ?

Accepted Answer

The correct answer is 1 16 . Let p and q represent the solutions to the given equation. Then, the given equation can be rewritten as 64 x - p x - q = 0 , or 64 x 2 - 64 p + q + p q = 0 . Since this equation is equivalent to the given equation, it follows that - 16 a + 4 b = - 64 p + q . Dividing both sides of this equation by - 64 yields 16 a + 4 b 64 = p + q , or 1 16 4 a + b = p + q . Therefore, the sum of the solutions to the given equation, p + q , is equal to 1 16 4 a + b . Since it's given that the sum of the solutions to the given equation is k 4 a + b , where k is a constant, it follows that k = 1 16 . Note that 1/16, .0625, 0.062, and 0.063 are examples of ways to enter a correct answer. Alternate approach: The given equation can be rewritten as 64 x 2 - 4 4 a + b x + a b = 0 , where a and b are positive constants. Dividing both sides of this equation by 4 yields 16 x 2 - 4 a + b x + a b 4 = 0 . The solutions for a quadratic equation in the form A x 2 + B x + C = 0 , where A , B , and C are constants, can be calculated using the quadratic formula, x = - B + B 2 - 4 A C 2 A and x = - B - B 2 - 4 A C 2 A . It follows that the sum of the solutions to a quadratic equation in the form A x 2 + B x + C = 0 is - B + B 2 - 4 A C 2 A + - B - B 2 - 4 A C 2 A , which can be rewritten as - B + - B + B 2 - 4 A C - B 2 - 4 A C 2 A , which is equivalent to - 2 B 2 A , or - B A . In the equation 16 x 2 - 4 a + b x + a b 4 = 0 , A = 16 , B = - 4 a + b , and C = a b 4 . Substituting 16 for A and - 4 a + b for B in - B A yields - - 4 a + b 16 , which can be rewritten as 1 16 4 a + b . Thus, the sum of the solutions to the given equation is 1 16 4 a + b . Since it's given that the sum of the solutions to the given equation is k 4 a + b , where k is a constant, it follows that k = 1 16 .

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