How Can We Use the Discriminant to Solve Quadratic Equations?

Understanding quadratic equations is really important in Year 11 Math. One key part of these equations is something called the discriminant. We write it as $D = b^2 - 4ac$. The discriminant helps us figure out the types of solutions (or roots) a quadratic equation has. This can be useful in real-life situations.

What is the Discriminant?

Let’s start with what a quadratic equation is. It usually looks like this:

$ax^2 + bx + c = 0$

In this equation, $a$, $b$, and $c$ are numbers, and $a$ can’t be zero. The discriminant $D$ helps us understand the solutions by telling us what kind of roots we have:

• If $D > 0$: There are two different real roots.
• If $D = 0$: There is one real root (it’s repeated).
• If $D < 0$: There are no real solutions, just complex roots.

How Does This Relate to Real-Life Problems?

When we have real-world problems, the discriminant helps us figure out what kind of answers we can expect. Let’s look at a couple of examples.

Example 1: Throwing a Ball

Imagine you are studying a ball thrown into the air. The height $h$ of the ball at any time $t$ can be shown with a quadratic equation like this:

$h(t) = -4.9t^2 + 20t + 1$

To find out when the ball hits the ground, we solve for $h(t) = 0$:

$-4.9t^2 + 20t + 1 = 0$

From this, we see that $a = -4.9$, $b = 20$, and $c = 1$. Now, let’s calculate the discriminant:

$D = 20^2 - 4(-4.9)(1) = 400 + 19.6 = 419.6$

Since $D > 0$, this means there are two different times when the ball will hit the ground.

Example 2: Area of a Triangle

Now imagine you need to find the base length of a triangle if you know the height and the area. The area $A$ of a triangle is shown with this formula:

$A = \frac{1}{2} \times \text{base} \times \text{height}$

If we rearrange that, we can find the base like this:

$\text{base} = \frac{2A}{\text{height}}$

Let’s say we want the area to be $A = 24$, but the height can change. We might get a quadratic equation like this:

$bh^2 - 48h + 24 = 0$

By checking the discriminant, we can see if the base length can be positive and real, or if it leads to a situation where we can't have a triangle at all.

Conclusion

The discriminant is a handy way to quickly check if solutions for quadratic equations make sense in different situations. Whether we’re looking at how a ball moves or properties of shapes, knowing how to evaluate $D$ helps us make smart choices in real life. Getting comfortable with this idea can help us understand quadratic functions better and see how they apply to the world around us. Remember, whether $D > 0$, $D = 0$, or $D < 0$, the discriminant is an important tool in math!