# High School Math Tip 13: Using the Discriminant of a Quadratic

The discriminant of a quadratic is very useful when finding the solution directly is not required and/or is too algebraic intensive.

To illustrate this, consider the following example:

Let the equation of a line be

for constants *m* and *b.*

Find the condition(s) on *m* and *b* such that the line is a tangent to a circle centred at the origin with radius 5.

To solve this, we know that the circle must thave equation

Solving this simultaneously with the equation of the line gives:

The solution to this equation gives us the *x*-coordinates of the point(s) of intersection between the line and the circle. In order for this line to be a tangent, there must be only one real solution. This means that the discriminant of this quadratic is zero:

Notice that this equation does not always give real solutions for *m* and *b.*

To ensure that we get real solutions for *m* and *b*, we observe that

Hence:

Essentially this means that in order for the line to be a tangent, it cannot be inside the circle, which makes sense.

Hence, as long as this condition on *b* is satisfied, solving for *m* using

will give us the equation of the tangent.

The discriminant is also useful in finding the maximum range for a projectile.

Let the launch velocity be *V,* the launch angle be *θ* and acceleration due to gravity be *g*.

Assuming that air resistance is negligible and that the projectile is launched from ground level, then the Cartesian equation of the projectile is:

The range is defined to be the horizontal displacement when the projectile hits the ground (*y*=0):

Since we are not interested in the initial horizontal displacement of zero, we can divide everything by *x:*

which usually has two real solutions for *θ*. But if the range is maximised, this *θ *must have only one real solution. Hence we require the discriminant to be zero:

since we are only interested in a positive horizontal displacement.

In order to find the optimal launch angle so that this maximum range is achieved, we can substitute this back into our quadratic equation for *θ*:

This technique can also be extended to cases where we are not launching from ground level.