Question

Let f(x) =x^2+Kx and g(x) = x+K. The graphs of f and g intersect at two distinct points. Find the value of K?

Answer 1

For graphs `f(x)` and `g(x)` to intersect at two distinct points, we must have `k!=-1`

Explanation:

As `f(x)=x^2+kx` and `g(x)=x+k`

and they will intersect where `f(x)=g(x)`

or `x^2+kx=x+k`

or `x^2+kx-x-k=0`

As this has two distinct solutions,

the discriminant of quadratic equation must be greater than `0` i.e.

`(k-1)^2-4xx(-k)>0`

or `(k-1)^2+4k>0`

or `(k+1)^2>0`

As `(k+1)^2` is always greater than `0` except when `k=-1`

Hence, for graphs `f(x)` and `g(x)` to intersect at two distinct points, we must have `k!=-1`