# Discriminants

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!