Background: This problem is from my textbook: Let $x^{2}-(m-3) x+m=0 (m \in R)$ be a quadratic equation. Find the values of $m$ for which at leat one one root lies in $(1,\ 2).$
The solution too this problem given in the book is like this:
Case I: Exactly one root lies in $(1,\ 2)$. So,
$$
f(1) f(2)<0 \Rightarrow m>10
$$
Case II: Both the roots lie in the interval $(1,\ 2)$. Then,
$
D \geq 0 \Rightarrow(m-1)(m-9) \geq 0
$
$$\Rightarrow \quad m \leq 1 \text{ or } m \geq 9.$$
$\text{also } f(1)>0$ and $f(2)>0$
$$\Rightarrow 10>m$$
$\text{and }1<-\frac{b}{2 a}<2$
$$\Rightarrow 5<m<7$$
Thus, no such $m$ exists.
Hence, $m \in(10, \infty)$
.
Question: But how did author decide which "properties" to use to solve such problems? In case I, exactly one roots lies in (1, \ 2), so the followings are the properties which are satisfied by such quadratic:
- The vertex of the quadratic is larger than $1$ but less than $2$.
- and Sum and products of the roots will be positive ie let.
- and the graph of the quadratic is concave up.
- and $f(1)f(2)\lt 0$
- and $af(2)\gt 0$
- and $af(1)\gt 0$
and so on… but the author just chose the last 3rd property, why? Surely we will get many other inequalities and equations, after "mathematicalizing" those properties, involving the term "$m$", shouldn't we solve all the inequalities and equation to get the solution; in the end there might be a chance that other equations/inequalities could have "limited" the set of possible values of $m$, that is we may get extraneous solutions to the problem.
[ Note the word "and", all these properties are satisfied, so suppose that after solving, say 1st property we get something like $m\lt 0$, then we will have to take intersection of this set and $m \in(10, \infty)$. ]
Similarly I can list many properties for second cases also and again the same problem.
So why is the author so sure that solving that specific inequality will give him solution set without any extraneous solutions? Indeed he did got it right. Surely there must be a way to know that after solving a specific equation/inequality generating from certain property will led you to a solution set without any extraneous solutions.
Best Answer
You have shown that "there exists a root in $(1,2)$" $\implies m > 10$. But this is not yet an equivalence, as you say there could be extraneous solutions introduced. However, note that the inverse is simple. We have: $m>10 \implies f(1)f(2) < 0 \implies$ "there exists a root in $(1,2)$".