Question

If the graph of the quadratic function f(x) = x^2 + dx +3d has its vertex on the x axis what are the possible values of d? What if f(x) = x^2 + 3dx +1?

Answer 1

The coordinates of vertex are: ( x , 0 ) and x = -b/2a
a ) For the 1st function: x = -d/2
0 = d² / 4 - d² / 2 + 3 d / · 4
0 = - d² + 12 d
d ( 12 - d ) = 0, d = 0 or d = 12
If d = 0, f (x) = x²
If d 0 12 f (x) = x² + 12 x + 36 = ( x + 6 )²
b ) For the 2nd function:
x = - b/2a = - 3 d/2
0 = 9 d² / 4 - 9 d / 2 + 1 / · 4
0 = 9 d² - 18 d + 4

`d_(12) = (18\pm √(324+144) )/(18)= (18\pm6 √(13) )/(18)`
d = 1 - √13/2 or d = 1 + √13/2