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
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.