y + k = x + 26 y - k = x 2 - 5 x In the given system

Question

y + k = x + 26 y - k = x 2 - 5 x In the given system of equations, k is a constant. The system has exactly one distinct real solution. What is the value of k ?

Accepted Answer

The correct answer is 35 2 . Subtracting the second equation from the first equation yields y + k - y - k = x + 26 - x 2 - 5 x , or 2 k = - x 2 + 6 x + 26 . This is equivalent to x 2 - 6 x + 2 k - 26 = 0 . It's given that the system has exactly one distinct real solution; therefore, this equation has exactly one distinct real solution. An equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants, has exactly one distinct real solution when the discriminant, b 2 - 4 a c , is equal to 0 . The equation x 2 - 6 x + ( 2 k - 26 ) = 0 is of this form, where a = 1 , b = - 6 , and c = 2 k - 26 . Substituting these values into the discriminant, b 2 - 4 a c , yields - 6 2 - 4 1 2 k - 26 . Setting the discriminant equal to 0 yields - 6 2 - 4 1 2 k - 26 = 0 , or - 8 k + 140 = 0 . Subtracting 140 from both sides of this equation yields - 8 k = - 140 . Dividing both sides of this equation by - 8 yields k = 35 2 . Note that 35/2 and 17.5 are examples of ways to enter a correct answer.

Question y-k-x-