To figure out what kind of roots a quadratic equation has, we can use something called the discriminant. This is a part of the quadratic formula.

A quadratic equation usually looks like this:

$ax^2 + bx + c = 0$

In this equation:

• a, b, and c are numbers.
• The discriminant (D) can be found using this formula:

$D = b^2 - 4ac$

The value of the discriminant helps us understand the roots:

1. If $D > 0$: There are two distinct real roots. This means the graph, which is a parabola, touches the x-axis at two different points.

   Example: For the equation $x^2 - 5x + 6 = 0$, we find:

   • $a = 1$, $b = -5$, and $c = 6$.
   • The discriminant is: $D = (-5)^2 - 4(1)(6) = 25 - 24 = 1$ Since $D$ is greater than 0, there are two different real roots.

2. If $D = 0$: There is exactly one real root, which is sometimes called a double root. In this case, the parabola just touches the x-axis at one point.

   Example: Look at the equation $x^2 - 4x + 4 = 0$.

   • Here, we calculate: $D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$ So, there is one double root.

3. If $D < 0$: This means the quadratic has no real roots. Instead, it has two complex roots. This tells us that the parabola does not touch the x-axis at all.

   Example: For the equation $x^2 + 2x + 5 = 0$, we find: $D = (2)^2 - 4(1)(5) = 4 - 20 = -16$ Since $D$ is less than 0, we know the roots are complex.

Using the discriminant is a simple way to figure out the type of roots in quadratic equations!