When you study complex numbers in Year 9 Mathematics, the quadratic formula can be really helpful!

The quadratic formula looks like this:

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

This formula helps us solve quadratic equations that look like $ax^2 + bx + c = 0$.

But what happens if the part inside the square root, called the discriminant ($b^2 - 4ac$), is negative? That’s when we get complex numbers as solutions!

Step-by-Step Example

Let’s check out an example: Solve the equation $x^2 + 4x + 8 = 0$.

1. Identify the numbers: In this equation, $a = 1$, $b = 4$, and $c = 8.

2. Calculate the discriminant: $b^2 - 4ac = 4^2 - 4(1)(8) = 16 - 32 = -16$

   Since the discriminant is negative, that means we will have complex solutions!

3. Use the quadratic formula: $x = \frac{-4 \pm \sqrt{-16}}{2(1)}$

4. Simplify the square root: Remember that $\sqrt{-16} = 4i$, where $i$ is called the imaginary unit.

5. Finish the calculations: $x = \frac{-4 \pm 4i}{2} = -2 \pm 2i$

So, the solutions are $x = -2 + 2i$ and $x = -2 - 2i$.

And that’s it! You’ve learned how to use the quadratic formula to find complex solutions!