Finding the values of $5a+b$ if $ax^2+bx+10$ does not have $2$ distinct roots If $ax^2+bx+10=0$ does not have two distinct real roots where( $a$ and $b$ are real) then the possible values of $5a+b$ from the given 4 options(as obviously there can be infinite possible values..but I only need to find out which of the four given possible values is one of these ...in short it is a Multi choice-multi-correct question)  are-( there can be more than one option correct)
1) -3
2) -2
3) -1
4) 0
my attempt
Here are some conclusions I have drawn-


*

*it is given that it does not have two distinct real roots so either( both roots are imaginary and conjugates of each other)- $D(b^2-4ac)<0$ (that is as $c=10$ so $b^2-40a<0$...or another case is both roots become equal that is $b^2-40a=0$.

*also I tried some manipulations like subtracting $f(3)-f(2)$ to get $5a+b$ ....now I think since it is given that the equation does not have two distinct real roots so $f(3)-f(2)$ or $5a+b$ cannot be equal to zero unless both $f(3),f(2)$ become equal to zero..which is not allowed (as real roots should not be distinct)so I think option (D) or zero shouldnt be possible but the answers given are $-2,-1,0$ (Edit:apparently $0$ is not the answer as pointed out be Tim phan below..but I still need help in proving it is equal to $-1,-2$)....also I am unable to get other answers I have tried getting $5a+b$ by keeping $x=5$ to get $5(5a+b+2)$ but I don't know whether it will be equal to zero or less than or greater than zero... As I don't know whether $5$ is a root of $f(x)$ or not..neither it is given in question...
Any help??
 A: If you write $5a + b = k$ and use this to get rid of the $a$ in the equation $b^2 - 40a \leq 0$, you get:
$$b^2 - 40\left(\frac{k - b}{5}\right) \leq 0$$
$$b^2 + 8b \leq 8k$$
Complete the square:
$$(b + 4)^2 \leq 8k + 16$$
$$(b + 4)^2 \leq 8(k + 2)$$
Since $b$ is arbitrary, the only restriction is that the left hand side is nonnegative. So you get:
$$0 \leq 8(k + 2)$$
$$-2 \leq k$$
Hence $k = 0, -1, -2$ are all possible.
A: The possible values (from those listed) are $0$, $-1$, and $-2$, obtained by $(a,b) = (0,0)$, $(1,-6)$, and $\left(\frac{2}{5},-4\right)$. (The first two are not unique, but the third one is; see below.)
Now, if $ax^2+bx+10$ does not have 2 distinct roots, then the discriminant is nonpositive, i.e. $b^2-40a\le 0$. Hence, $40a\ge b^2$, necessarily implying that $a\ge 0$. Furthermore, 
$$b^2\le 40a\implies |b|\le\sqrt{40a}\implies b\ge -\sqrt{40a}$$
and hence $5a+b \ge 5a-\sqrt{40a}$. It can be checked that
$$\min\limits_{a\ge 0}{\left(5a-\sqrt{40a}\right)} = -2 $$
achieved at $a = \frac{2}{5}$. Hence, if $ax^2+bx+10$ does not have 2 distinct roots, then
$$5a+b\ge -2$$
with equality if and only if $a = \frac{2}{5}$ and $b = -\sqrt{40a} = -4$.
A: Assume $b + 5a = 0$, then $b = -5a$ and $a = -\frac{1}{5}b$. Also, we have that $b^2 < 4ac$. Combine these together to obtain $b^2 < -\frac{4}{5}b c$ and $25a^2 < 4a c$.
(1) If $a>0, b>0$, then we have: $b<-\frac{4}{5}c$< which implies $c<0$. False.
(2) If $a<0, b<0$, then we have: $-25|a|>4c$ which implies $c<0$. False.
If $a>0, b<0$, you have the same thing as in (1) and if If $a<0, b>0$, you have the same thing as in (2). Thus it is not possible for $b + 5a = 0$.
Proceed similarly.

Second part:
let $D>0$, assume $5a + b =-D$, so $b = -(D+5a)$. Then for this equation to not have two distinct real roots: $b^2 \leq 40a$. replacing $a$ in to obtain:
$ (D+5a)^2 < 40a$ Notice that this implies $a > 0$. Simplify:
$$D + 5a < \pm\sqrt{40 a}$$
Let $u = \sqrt{a}$, then we have $5u^2 \mp \sqrt{40}u + D < 0$ is the condition, which is if we can find such $u$ under some condition on $D$, then the same condition is also the condition for $ax^2 + bx + 10 = 0$ to not have two distinct real roots.
Solve this to get 
$$u=\frac{1}{10}\left(\sqrt{40} - \sqrt{40 - 20 D}\right)$$
this has a solution if $D \leq 2$, which means if $D \leq 2$, then the $ax^2 + bx + 10 = 0$ does not have two distinct real roots. Thus the possible solutions are $5a + b = -1, -2$, or (3) and (2).
Please check my algebra carefully.
A: I found that if a = 1/5 and b = -1 then 5a+b = 0. 
Did i break any rules?
