The Discriminant

What are roots and the discriminant?

The discriminant of a quadratic equation will tell you how many roots the quadratic equation has. For example, in physics, while analyzing projectile motion, one can use the discriminant to predict the number of times the projectile will hit the ground – essentially the roots of the equation. This principal notion is essential not only in quadratic equations but also in higher-order polynomials. Solutions of a quadratic equation are known as roots; they can be seen on the graph of a quadratic where the graph crosses the x-axis. Recall that the general quadratic equation: ax^2+bx+c=0, can have two distinct roots, one repeated root or no roots at all (unless you are working with complex numbers).

Here are three examples that illustrate how quadratics can have different numbers of roots:

Note that we use a negative quadratic to illustrate that a quadratic may have no roots but it is also possible for a positive quadratic to have no roots as well.

The Value of the Discriminant

The discriminant can tell you how many roots a quadratic equation will have without having to actually find them. For the quadratic equation ax^2+bx+c=0, the discriminant is given by b^2-4ac.

Firstly, if b^2-4ac\hspace{2pt}>\hspace{2pt} 0, the equation has two distinct roots.

Alternatively, if b^2-4ac = 0, the equation has one repeated root.

Lastly, if b^2-4ac\hspace{2pt}<\hspace{2pt}0, the equation has no roots.

See Examples and see discriminants for higher order polynomials

Examples

Given that the quadratic equation kx^2-4x+2 has equal roots, find the value of k.

Solution:

If a quadratic equation has equal roots then b^2-4ac=0 where a, b and c are identified from the quadratic ax^2+bx+c. In this case, a=k, b=-4 and c=2. The discriminant is then (-4)^2-4\times k\times 2=16-8k. If this must be zero then the value of k must be 2.

A quadratic has equation y=px^2+3px-5.
Find an expression, in terms of p, for the discriminant of the quadratic.
Given that the quadratic has two distinct roots, find the possible values of p.

Solution:

  1. The discriminant is b^2-4ac where a=p, b=3p and c=-5. It follows that it is (3p)^2-4\times p\times-5=9p^2+20p.
  2. If the quadratic has two distinct roots, the discriminant must be positive, i.e. 9p^2+20p\hspace{2pt}>\hspace{2pt}0 or p(9p+20)\hspace{2pt}>\hspace{2pt}0. In order for this inequality to hold we must have p\hspace{2pt}>\hspace{2pt} 0 or p\hspace{2pt}<\hspace{2pt}-\frac{20}{9} (you can see this by sketching 9p^2+20p and seeing where it is positive – see Inequalities).

Videos

https://youtu.be/LsQfA3hCm7Q

Finding the discriminant of a quadratic with algebraic coefficients and using the fact there are no roots to find the values for the unknown value.