Quadratic equations are a really interesting topic! They are an important part of algebra, and they can lead us on all sorts of math adventures! Today, we’ll look at examples of quadratic equations that have only complex roots. How cool is that? Let’s jump in and learn how to spot these fascinating equations!

What is a Quadratic Equation?

A quadratic equation usually looks like this:

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

In this equation, $a$, $b$, and $c$ are numbers, and $a$ cannot be zero. Quadratic equations can have different types of roots—real roots and complex roots. It’s important to understand these roots, and we can figure them out using something called the discriminant!

What is the Discriminant?

The discriminant helps us understand the roots of the equation. We find it using this formula:

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

Where:

• $D$ is the discriminant,
• $b$ is the number in front of $x$,
• $a$ is the number in front of $x^2$, and
• $c$ is the constant term (the number without a variable).

Identifying Complex Roots

So, how can we tell if a quadratic equation has complex roots? It’s pretty straightforward! The quadratic formula gives us these clues:

• If $D > 0$: the equation has two different real roots.
• If $D = 0$: the equation has one real root (which is counted twice).
• If $D < 0$: the equation has two complex roots!

Yay! Now let’s check out some examples of quadratic equations with complex roots.

Examples of Quadratic Equations with Complex Roots

1. Example 1: Let’s look at this equation:

   $$x^2 + 4x + 8 = 0$$

   Here, $a = 1$, $b = 4$, and $c = 8$. Now, let’s calculate the discriminant:

   $$D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16$$

   Since $D < 0$, this equation has complex roots!

2. Example 2: Here’s another example:

   $$2x^2 + 2x + 5 = 0$$

   For this one, $a = 2$, $b = 2$, and $c = 5$. Now, we’ll calculate the discriminant:

   $$D = 2^2 - 4 \cdot 2 \cdot 5 = 4 - 40 = -36$$

   Again, since $D < 0$, this equation also has complex roots!

3. Example 3: Let’s try this one:

   $$x^2 - 2x + 3 = 0$$

   In this case, $a = 1$, $b = -2$, and $c = 3$. Calculating the discriminant gives us:

   $$D = (-2)^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8$$

   Once more, since $D < 0$, this equation has complex roots!

Conclusion

Isn’t it amazing that we can use the discriminant to figure out the types of roots in quadratic equations? By noticing when $D < 0$, we can discover a whole new world of complex numbers hiding within these equations! So, keep practicing and have fun with these wonderful equations! Happy studying!