THIS ONE REALLY DONT KNOW
For what value(s) of a the expression
y = ax² +10x + 25 is never positive for all
x belongs to R.
recall the function of the discriminant -- it tells you how many roots there are, and what kind. Here, the discriminant is
b^2 - 4ac = 100 - 100a
If y is never positive, then it must mean that it has a single root, or it has no root, and opens downward.
The discriminant is zero (1 root) if a=1, but that means the parabola opens upward, so that will not work.
Otherwise, if a>1 then there are no real roots, but the parabola still opens upward.
So no value of a will leave y with no positive values.
After all,
ax^2+10x + 25
= a(x + 10/a) + 25
= a(x + 5/a)^2 + 25 - 25/a
But we know that for the parabola to open downward, a<0 but that means that 25 - 25/a is positive -- and that is the y-value at the vertex.