What will be the sign of the constant term ‘c’?

Question

We have a quadratic equation $ax^2$+ $bx$ + c =0 . It has no real roots. i.e $b^2$- $4ac$ < 0 .What is the sign of c ?

My tries- I have supposed the $4ac$ will be greater than $b^2$ , because if it wasn’t then the roots will never be less then zero . So Two cases arise-

1- Both $a$ and $c$ are greater then zero.

2- Both $a$ and $c$ are smaller then zero.

So $c$ could be greater than zero or smaller than zero .

But my book says, $c<0$ . WHY?

Answer 1

Two cases :
Your equation is $(a+b)x^2=-c$.(I suppose you have a typo ).This does not tell us anything about the sign of $c$.
Your equation is $ax^2+bx+c=0$ .This does not tell us anything about the sign of $c$.
But what you can say is that the sign of $c$ is the same with $a$ in order to have $D<0$ .

Answer 2

As you have correctly noted in the original post, what matters is not the sign of $c$, but rather the sign of $b^2-4ac$. Here are a few possibilities:

1. If $a$ and $c$ have opposite signs, then $4ac$ is negative, and hence $b^2-4ac$ will be positive, so the equation will have two real solutions.
2. Therefore if the equation has no real solutions, then $a$ and $c$ must have the same sign.
3. However, while the condition that $a$ and $c$ have the same sign is necessary, it is not sufficient to ensure that there are no real solutions. In order to ensure that there are no solutions, $a$ and $c$ have to have the same sign and have a product that is larger than $b^2$.