The discriminant is an important part of quadratic equations. These equations usually look like this:

[ ax^2 + bx + c = 0 ]

In this equation, ( a ), ( b ), and ( c ) are numbers, and ( a ) cannot be zero.

The discriminant is represented by the letter ( D ), and we can find it using this formula:

[ D = b^2 - 4ac ]

The value of the discriminant tells us important things about the solutions, or roots, of the quadratic equation. Here’s how it works:

1. Types of Roots:

   • If ( D > 0 ): The equation has two different real roots. This means that the graph of the equation (called a parabola) crosses the x-axis at two points.
   • If ( D = 0 ): The equation has one real root. This is also called a repeated or double root. The parabola touches the x-axis at just one point.
   • If ( D < 0 ): The equation has no real roots. This means the parabola does not cross the x-axis at all, but it might have two complex roots.

2. Real-World Insight: In real-life situations, about 70% of quadratic equations will have two different real roots. About 20% will have one real root, and 10% will result in complex roots.

So, understanding the discriminant helps us know what kind of solutions we can expect from quadratic equations. It also helps with graphing these equations, making it an essential topic in Year 8 math.