AS-Level Maths Question

I'm stuck on the following question :< please help <3

Prove that, for all values of x, x² + 6x + 18 > 2 - 1/2x

Try moving it all to one side and then complete the square. Or alternatively differentiate to find the minimum of the quadratic, if you know how to do that.

What do you do after you’ve found the minimum point?

You'd like to show that it lies above the line $2-\frac{1}{2}x$. Though this is not sufficient to prove the statement. In addition to this, you want to show that the quadratic on the LHS and the line on the RHS do not intersect, so that's just a simple case of applying the discriminant and showing that it is <0.

It would be much more straight-forward to just move the -1/2x to the LHS to get $x^2+6.5x+18>2$. Now here, it is indeed sufficient to show that the minimum point of the quadratic on the left is above the line y=2.