Help quadratic equation questions

Question

Determine the value of c if y=3x2+6x-c only has 1 x‒intercept. 

Prove algebraically.

Answer 1

3x²+6x-c must have only one root, so 3(x²+2x-c/3) must contain a perfect square.

Therefore x²+2x-c/3 is a perfect square. This can be written x²+2x+1-1-c/3=(x+1)²-1-c/3.

And -1-c/3=0 so that the single root can be x=-1. Therefore c/3=-1, c=-3.

The quadratic becomes 3x²+6x+3=3(x+1)².