y = 2 x 2 - 21 x + 64 y = 3 x + a In the given system

Question

y = 2 x 2 - 21 x + 64 y = 3 x + a In the given system of equations, a is a constant. The graphs of the equations in the given system intersect at exactly one point, ( x , y ) , in the xy -plane. What is the value of x ?

Accepted Answer

Choice C is correct. It's given that the graphs of the equations in the given system intersect at exactly one point, x , y , in the xy -plane. Therefore, x , y is the only solution to the given system of equations. The given system of equations can be solved by subtracting the second equation, y = 3 x + a , from the first equation, y = 2 x 2 - 21 x + 64 . This yields y - y = 2 x 2 - 21 x + 64 - 3 x + a , or 0 = 2 x 2 - 24 x + 64 - a . Since the given system has only one solution, this equation has only one solution. A quadratic equation in the form r x 2 + s x + t = 0 , where r , s , and t are constants, has one solution if and only if the discriminant, s 2 - 4 r t , is equal to zero. Substituting 2 for r , - 24 for s , and - a + 64 for t in the expression s 2 - 4 r t yields - 24 2 - 4 2 64 - a . Setting this expression equal to zero yields - 24 2 - 4 2 64 - a = 0 , or 8 a + 64 = 0 . Subtracting 64 from both sides of this equation yields 8 a = - 64 . Dividing both sides of this equation by 8 yields a = - 8 . Substituting - 8 for a in the equation 0 = 2 x 2 - 24 x + 64 - a yields 0 = 2 x 2 - 24 x + 64 + 8 , or 0 = 2 x 2 - 24 x + 72 . Factoring 2 from the right-hand side of this equation yields 0 = 2 x 2 - 12 x + 36 . Dividing both sides of this equation by 2 yields 0 = x 2 - 12 x + 36 , which is equivalent to 0 = x - 6 x - 6 , or 0 = x - 6 2 . Taking the square root of both sides of this equation yields 0 = x - 6 . Adding 6 to both sides of this equation yields x = 6 . Choice A is incorrect. This is the value of a , not x . Choice B is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Question y-2-x-