There is a question in my maths exercises textbook that I have got a different answer than the one given in my textbook.
The question is :-
For the following graph of the quadratic equation $$ y = ax^2 + bx + c $$ is the product abc is negative?
So my answer is:-
In the figure (a), the parabola is downward so $$ a < 0 $$ Since the parabola intersects the y-axis at a negative point, $$ c < 0$$ Since the x-vertex is negative $$ \frac{-b}{2a} < 0 $$ $$ -b < 0$$ $$ b > 0 $$ As 'a' is negative, 'b' is positive and 'c' is negative, their product abc must be positive. However the textbook says that the product is negative. Is it a printing mistake or is my answer wrong? Thanks in advance.
The only error in your reasoning is when you said $a,c<0$ and $\frac{-b}{2a}<0$ implies $-b<0$ - it should be $-b>0$, from which $b<0$. Then $a,b,c<0$ implies $abc<0$.
When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips.