To understand how the discriminant affects the roots of quadratic equations, it’s important to look at the relationships between the discriminant value and the type of roots using graphs.

A quadratic equation can be written like this:

$$ax^2 + bx + c = 0$$

The discriminant (we often call it (D)) is found using this formula:

$$D = b^2 - 4ac$$

What the Discriminant Tells Us

The value of the discriminant tells us how many solutions (or roots) there are for the quadratic equation. Here are the different cases:

1. Two Real and Different Roots: This happens when (D > 0). In this case, the graph of the equation curves and crosses the x-axis at two different points.

   • For example, if we take the equation (x^2 - 5x + 6 = 0), we have (a = 1), (b = -5), and (c = 6). Calculating the discriminant gives us: $$D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0$$
   • When we graph (y = x^2 - 5x + 6), we see that it crosses the x-axis at (x = 2) and (x = 3).

2. One Real Root (Repeated): This occurs when (D = 0). Here, the graph just touches the x-axis at one point.

   • For the equation (x^2 - 4x + 4 = 0), we have (a = 1), (b = -4), and (c = 4). The discriminant is calculated like this: $$D = (-4)^2 - 4(1)(4) = 16 - 16 = 0$$
   • Graphing (y = x^2 - 4x + 4) shows it just touches the x-axis at (x = 2). This means there is one repeated root at (x = 2).

3. No Real Roots (Two Complex Roots): This happens when (D < 0). In this case, the graph does not touch the x-axis at all.

   • For example, in the equation (x^2 + 2x + 5 = 0), we have (a = 1), (b = 2), and (c = 5). The discriminant calculation looks like this: $$D = (2)^2 - 4(1)(5) = 4 - 20 = -16 < 0$$
   • When we graph (y = x^2 + 2x + 5), we see that the curve opens upwards and never touches the x-axis. This means the roots are complex.

Visualizing the Discriminant

It can really help to see how the discriminant changes the graph of the quadratic:

• When (D > 0): The graph intersects the x-axis at two points.
• When (D = 0): The graph touches the x-axis at one point (the vertex).
• When (D < 0): The graph does not touch the x-axis at all, meaning the roots are complex.

In Summary

By looking at the discriminant of a quadratic equation, you can tell what kind of roots there are without even solving it. The graphs help show these ideas clearly:

• (D > 0): Two different real roots.
• (D = 0): One repeated real root.
• (D < 0): Two complex roots.

Understanding this relationship is very important in grade 10 algebra. It helps build a strong foundation for future math studies and makes solving problems easier.