Relationship between Discriminant and coefficients of a Quadratic function

Question

To answer question (b), the problem requires me to use the discriminant. However, I still do not understand how would I realize that I would have to make use of it. Can someone please explain the relationship between both inequalities?

P.S. For those who think I am looking for answers to a homework, I have the answer to (b) so there is no need to provide it.

Answer 1

Recall that a quadratic equation has the form $ax^2 + bx + c = 0$, $a \neq 0$. The solutions are given by the quadratic formula $$x = \frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$$

The critical part of this formula is the part under the square root, $b^2 - 4ac$. We define this quantity to be the discriminant, which generally has the symbol $\Delta$.

So $\Delta = b^2 - 4ac$. There are three cases:

1. $b^2 - 4ac > 0$. In this case, our equation will have two distinct real solutions.
2. $b^2 - 4ac = 0$. In this case, one of our solution will be $+0$ and the other will be $-0$, but hopefully you can see that adding and subtracting $0$ will just give the same solution. So in this case, we have two equal roots (or just one distinct root).
3. $b^2 - 4ac < 0$. In this case, we will be taking the square root of a negative number. This is not defined for real numbers, so we say that our equation has no real roots in this case.

Now let's look at your problem.

Part (a). The answer is just $f'(x) = 3px^2 +2px +q$.

Part (b). We are told that $f'(x) \geq 0$. This means that when we plot the graph $y=f'(x)$, the graph does not go below the $x$ axis. It must either touch the $x$ axis or be completely above the $x$ axis. This corresponds to either $f'(x)=0$ either having a repeated root or having no real solutions.

Hence, we must be in case 2 OR case 3, with $b^2 - 4ac = 0$ or $b^2 - 4ac < 0$. Both of these cases are covered by just requiring $b^2 - 4ac \leq 0$.

For our equation, $a = 3p$, $b=2p$ and $c=q$, so we just substitute this in to obtain: $$(2p)^2 - 4(3p)(q) \leq 0 $$ $$\Rightarrow 4p^2 - 12pq \leq 0$$ $$\Rightarrow 4p^2 \leq 12pq$$ and dividing both sides by 4 we get the result $$p^2 \leq 3pq$$

Generally speaking, with these types of quadratic questions, you are looking to identify which of the three cases your equation is in and use that obtain an inequality involving the coefficients.

Answer 2

Recall that for a quadratic $g(x) = ax^2 + bx + c$, and its discriminant $\Delta = b^2-4ac$.

If $\Delta$ is negative, then $g(x)$ has no real roots, and is either always positive or always negative for all real $x$.

If $\Delta$ is zero, then $g(x)$ has a double root, but is still either positive only or negative only for other real $x$.

So if given that $g(x) \ge 0$ for all real $x$, then:

• the discriminant is either zero or negative: $\Delta = b^2 - 4ac \le 0$; and
• the coefficient of $x^2$ is positive: $a > 0$.

Otherwise, by having positive $\Delta$, $g(x)$ will give positive value for some $x$ and negative value for some other $x$.