The most general expression of a quadratic equation is shown below:

where \(a\), \(b\) and \(c\) are real *constants*, with \(a\neq 0\). For instance, the following equation:

is a quadratic equation, whereas

is not (because the factor \(x^2\) 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 \(a\), \(b\) and \(c\). In this case, \(a=2\), \(b = -1\) and \(c = -1\):

Now, we see that we have two solutions because of the \(\pm\), which means that the roots are

**The discriminant**

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 *Discriminant*:

The discriminant can be negative, zero or positive, and the type of solutions will depend on it. In fact, we have that

- If \(D > 0\): There are two different real roots
- If \(D = 0\): There only one real root (the roots are repeated)
- If \(D < 0\): 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
\( \sqrt{ b^2-4ac}\) appears, which won't be real if \(b^2-4ac <0\). To see graphically how to locate the
roots, you could try the our quadratic equation solver

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