Explanation:
your text cuts off. so, I will answer a).
first, let's write things in a nicer and clearer way :
f(x) = 2x² + 12x + 18
right ?
the discriminate we are talking about is
b² - 4ac
remember, for a quadratic equation
ax² + bx + c = 0
the general solution for x is
x = (-b ± sqrt(b² - 4ac))/(2a)
so, the mentioned discriminate is the content of the square root function in the above expression.
and it is the only critical part to decide if there are 2 real solutions (roots), one real solution or 2 complex solutions.
remember, a square root only has real solutions, if the content is positive. if the content is negative, we have complex solutions. if the content is 0, we have only one solution.
in our case
a = 2
b = 12
c = 18
so,
b² - 4ac = 12² - 4×2×18 = 144 - 144 = 0
as explained, that means there is only one real solution.
but if it helps to better understand : it is counting as 2 solutions. it just means that the vertex of the parabola is on the x-axis, and the parabola is just touching the x-axis there, and is not intersecting it.
if we shift the parabola down just a tiny bit (by decreasing c a tiny bit), we would suddenly get 2 solutions as normally expected.
and if we shift it up just a tiny bit (by increasing c a tiny bit), we would lose the "touch" on the x-axis and get only 2 complex solutions.