My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was
Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ .
My teacher said that there are no solutions for this function when it is greater than $0$ and used $b^2-4ac \lt 0$, and this is the part that confused me. I understand why he used $b^2-4ac \lt 0$ but I cannot understand why there are no solutions by just looking at the function. Could someone explain this to me?
Best Answer
Edit: I will try to say something general first. Hopefully that will make my answer a bit easier to understand. But as a first thing, it might be helpful to take a look at for example the Wikipedia article on quadratic equations.
If you have a function $f(t)$, then $t$ is a variable (called the independent variable). When we change the value of $t$, then the value of $f(t)$ changes. For example if $f(t) = t +2$, then for $t = 1, f(t) = f(1) = 3$. For $t = 2, f(t) = t(2) = 4$.
Now given a function $f(t)$, you can ask the question: does there exist a value or $t$ such that for that value $f(t)$ is zero. You are asking if there are any zeros. Anything that makes that function equal to zero.
A special type of function are the quadratic polynomials. Here $f(t)$ looks like this:
$$ f(t) = at^2 + bt + c. $$ ($a,b,c$ are fixed constants.) For example $f(t) = t^2 + 3t + 2$. Now we ask the question: does there exist a number such that the value of $f(t)$ is $0$? The answer is yes because $$f(-1) = (-1)^2 + 3\cdot(-1) + 2 = 0.$$
Now if you know about a function that $f(t)>0$ for all $t$, then no matter what $t$ is $f(t)$ is never going to be equal to zero. It is always positive. That is exactly what it means to say that $f(t) > 0$ for all $t$.
Another way to answer the question about whether a function is ever equal to zero is to draw the graph of the function. If the function intersects the $t$-axis (think: $x$-axis) then there is some number for which the value of the function is zero. That is exactly what it means to intersect the $t$-axis.
So to echo what other people have already mentioned: you are considering a polynomial: $$ P(t) = at^2 +5t + 4. $$ If you sketch the graph of this polynomial you get a parabola. The question can be asked whether the graph of the polynomial intersects the $t$-axis. Asking this question is the same as asking for roots (or zeros) of the polynomial. I.e. you want to know whether of not there is a number $t_0$ such that $P(t_0) = 0$.
You are assuming (supposing) that $P(t) > 0$ for all $t$. I.e. that means that there are no roots of the polynomial, i.e. no solution to the equation $P(t) =0$. First you might notice that if $P(t) >0$ for all $t$, then $a$ has to be a positive number. If $a$ was negative, then for $t = -100000$, $P(t)$ would be negative, right?
So what more can we say? We have the discriminant that tells you something about how many roots the polynomial has. For a general quadratic equation: $at^2 + bt + c= 0$ ($a,b,c$ being constants here and $t$ the variable), the discriminant is defined as $$ d = b^2 - 4ac. $$
It is a fact that if $$ \begin{align} d &> 0\quad \text{ means that there are exactly two distinct roots}\\ d &= 0\quad \text{ means that there are exactly one root}\\ d &< 0\quad \text{ means that there are no roots.} \end{align} $$
For your polynomial, you would then need $d = 5^2 - 4a\cdot 4 < 0$. So $$ \begin{align} 5^2 - 4a\cdot 4 &< 0 &\Rightarrow\\ - 4a\cdot 4 &< -25 &\Rightarrow\\ a &> \frac{25}{16}. \end{align} $$ And there you go.