Putting the fancy name aside, a discriminant is the \(b^2-4ac\) in \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) (i.e. the quadratic equation).

It turns out that the value of this expression can tell us about the number of intersections a quadratic equation will have with a straight line on a graph.

- if the discriminant > 0, then there are 2 intersections (or solutions, if you will)
- if the discriminant = 0, then there there is 1 solution (i.e. the straight line is a tangent to the curve)
- if the discriminant < 0, then there is no solution

If, for whatever reason, we feel the urge to know how many intersections the line \(y=3x^2-7x+9\) will have with \(y=0\) (the \(x\) axis), then we can simply calculate the value of its discriminant.

if \(a=3, b=-7, c=9\), then the discriminant is \((-7)^2-4(3)(9)\) which yields -59. Looking at the rules above, we can satisfy our urge to know the number of intersections this line has with \(y=0\): zero.

Similarly, we can use this information to find any missing values, or a possible value, in a quadratic if we know the number of intersections. Take \(5x^2-3x+k=0\), which supposedly has a tangent that is the \(x\) axis. We need the value of \(k\).

We know that for a tangent to exist, our discriminant has to be 0. Hence,

\((-3)^2-4(5)(k)=0\)

Solve as normal,

\(k=\frac{9}{20}\)

Now, let's find the value of \(k\) if \(x^2+y^2=50\) and \(x=2y+k\) have two points of intersection.

\(x\) is already the subject in the second equation, so let's put that into the first and solve from there.

\((2y+k)^2+y^2=50\)

\(4y^2+4ky+k^2+y^2=50\)

\(5y^2+4ky+k^2-50=0\) (relate this to \(ax^2+bx+c=0\))

Now let's work on the discriminant. We know it's greater than zero, since there are two solutions. Hence, we'll start working with inequalities.

\((4k)^2-4(5)(k^2-50)>0\)

\(16k^2-20k^2+1000>0\)

\(-4k^2>-1000\)

\(k^2\gt250\)

Let \(k^2=250\)

\(k=\pm5\sqrt{10}\)

This tells us that \(k\) lies somewhere in between \(-5\sqrt{10}\lt k\lt5\sqrt{10}\).

That's enough.

This article was written on 27/09/2023. If you have any thoughts, feel free to send me an email with them. Have a nice day!