When we're learning about quadratic equations in Grade 10 Algebra, one important idea we come across is something called the discriminant. The discriminant is part of the quadratic formula, which looks like this:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

In this formula, the part $b^2 - 4ac$ is called the discriminant. It gives us useful information about the roots (or solutions) of the quadratic equation. Quadratic equations are usually written in the form $ax^2 + bx + c = 0$.

What If the Discriminant Is Zero?

When the discriminant ($D$) is zero, it means the quadratic equation has one root that is special. This is often called a repeated root or a double root.

This is important because it tells us that the parabola, which we get from the quadratic equation, just touches the x-axis at one point. Instead of crossing over the x-axis, it only "kisses" it!

Why Is This Important?

1. Understanding the Graph:

   • Imagine a U-shaped curve (this is the parabola). If the discriminant is positive ($D > 0$), the curve crosses the x-axis at two points. This means there are two different real roots.
   • If the discriminant is negative ($D < 0$), the curve stays above or below the x-axis. This means there are no real roots at all!
   • But when $D = 0$, the very top point (called the vertex) of the parabola sits right on the x-axis. This can be shown like this: $y = a(x - r)^2$ where $r$ is that repeated root.

2. Real-Life Examples:

   • There are many real-world situations where there is only one possible solution. For example, think about a ball that is thrown up into the air. It reaches its highest point and then just barely touches the ground before bouncing back up. This would relate to a quadratic equation with a zero discriminant.

Example:

Let’s look at the quadratic equation $x^2 - 6x + 9 = 0$. Here, the numbers (called coefficients) are $a = 1$, $b = -6$, and $c = 9$.

• First, we will find the discriminant:

  $D = b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0$

Since $D = 0$, we have one double root.

• Now, we can use the quadratic formula to find it:

  $x = \frac{-(-6) \pm \sqrt{0}}{2(1)} = \frac{6}{2} = 3$

So, the root $x = 3$ is repeated.

In conclusion, when a quadratic equation has a zero discriminant, it’s significant because it shows that the parabola just touches the x-axis at one spot. This idea helps us understand how quadratic functions behave both in math problems and in real life!