The curve y=(k-6)x^2 - 8x + k cuts the x axis at two points and has a minimum point. Find the range of k. [Answer: 6<k<8]

Question

Pls help me with the working

Answer 1

The graph is a parabola.  Since the graph has a minimum point, k-6 > 0.  So, k > 6.

 

Since the graph has 2 x-intercepts, the discriminant must be positive.  

 

    So, (-8)² - 4(k-6)(k) > 0

 

          64 - 4k(k-6) > 0

 

          -4k² + 24k + 64 > 0

 

           -4(k² - 6k - 16) > 0

 

            k² - 6k - 16 < 0

 

            (k-8)(k+2) < 0

 

            Case 1:  k-8 > 0 and k + 2 < 0

 

                         Then, k > 8 and k < -2  (not possible)

 

            Case 2:  k-8 < 0 and k+2 > 0

 

                         Then, k < 8 and k > -2

 

So, k < 8, k > -2, and k > 6.

 

For all three inequalities to be true, we must have 6 < k < 8.