Hello sym0110,

for delta, you can use the \Delta command :

For the second question, you must show that for any

, the equation

has at least one real solution in

. Check the discriminant of this quadratic equation to see if it can actually have real solutions (negative discriminant means no real solution). Can you follow up ?

Take your quadratic equation :

.

This can be factorized as :

.

Take the discriminant :

.

We then have :

.

We want to prove that

, so

.

That is :

Simplify further :

Keep going :

Is this correct ? Let us check the discriminant of this new quadratic :

.

If the discriminant is negative, it means the equation has no real roots and thus all values are greater than zero (this is because the

value of the quadratic equation is positive. If it was negative, then all values would be less than zero. Graph some equations to see what I mean). Thus we have proved that :

And therefore we have proved that for any real

, the line intersects the curve (finish the conclusion properly).