57 x 2 + ( 57 b + a ) x + a b = 0 In the given equation,

Question

57 x 2 + ( 57 b + a ) x + a b = 0 In the given equation, a and b are positive constants. The product of the solutions to the given equation is k a b , where k is a constant. What is the value of k ?

Accepted Answer

Choice A is correct. The left-hand side of the given equation is the expression 57 x 2 + 57 b + a x + a b . Applying the distributive property to this expression yields 57 x 2 + 57 b x + a x + a b . Since the first two terms of this expression have a common factor of 57 x and the last two terms of this expression have a common factor of a , this expression can be rewritten as 57 x x + b + a x + b . Since the two terms of this expression have a common factor of x + b , it can be rewritten as x + b 57 x + a . Therefore, the given equation can be rewritten as x + b 57 x + a = 0 . By the zero product property, it follows that x + b = 0 or 57 x + a = 0 . Subtracting b from both sides of the equation x + b = 0 yields x = - b . Subtracting a from both sides of the equation 57 x + a = 0 yields 57 x = - a . Dividing both sides of this equation by 57 yields x = - a 57 . Therefore, the solutions to the given equation are - b and - a 57 . It follows that the product of the solutions of the given equation is - b - a 57 , or a b 57 . It&rsquo;s given that the product of the solutions of the given equation is k a b . It follows that a b 57 = k a b , which can also be written as a b 1 57 = a b k . It&rsquo;s given that a and b are positive constants. Therefore, dividing both sides of the equation a b 1 57 = a b k by a b yields 1 57 = k . Thus, the value of k is 1 57 . Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Question 57-x-2