Algebra Tutorials - Quadratic Equations
The most general expression of a quadratic equation is shown below:
where , and are real constants, with . For instance, the following equation:
is a quadratic equation, whereas
is not (because the factor is not present in the equation).
Solving the Quadratic Equation
The main objective when we have a quadratic equation is to find its solutions or roots, the other
name that is commonly used. The roots are computed with the well known quadratic formula
Example: Find the roots of the equation
Solution: We need to apply the quadratic equation formula, and replace the corresponding
values of ,
. In this case,
Now, we see that we have two solutions because of the ,
which means that the roots are
It turns out that we can know a lot about the roots of a quadratic equation before even solving it. How is that possible?
Well, we need to compute the following quantity, which is called the Discriminate:
The discriminate can be negative, zero or positive, and the type of solutions will depend on it. In fact, we have that
- If : There are two different real roots
- If : There only one real root (the roots are repeated)
- If : There are no real roots (The roots are complex)
So, depending on the value of the discriminate we'll be able to determine beforehand what kind of solutions.
Why we get
complex roots with a negative discriminate? Well, because in the quadratic formula, the term
appears, which won't be real
if . To see graphically how to locate the
roots, you could try the quadratic equation solver.