Completing the square

Question

The function f(x) = x² - 8kx + 20k² + 1 has a minimum value of r² + 4k.

 

Using completing the square method, show that r = 2k - 1

Answer 1

f(x) = x²-(8k)x + (20k²+1) 

 

      = x² - (8k)x + 16k² - 16k² + 20k² + 1

 

      = (x-4k)² + 4k² + 1

 

The vertex (lowest point) = (4k, 4k²+1)

 

So, 4k²+1 = r²+4k

 

           r² = 4k²-4k+1

 

           r² = (2k-1)²

 

           r = ±(2k-1)

 

I get two possibilities for r:  r = 2k-1 or r = 1-2k.