When we solve quadratic equations, we sometimes find solutions that are complex. This happens when a part called the discriminant is negative.
The discriminant is figured out using this formula:
[ D = b^2 - 4ac ]
Let’s look at an example to understand this better.
Example 1:
For the equation ( x^2 + 4x + 8 = 0 ), we calculate the discriminant like this:
[ D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16 ]
Here, ( D ) is less than zero, which means we have complex solutions.
Finding the Solutions:
We use the quadratic formula to find the solutions. The formula is:
[ x = \frac{-b \pm \sqrt{D}}{2a} ]
Plugging in our numbers gives us:
[ x = \frac{-4 \pm \sqrt{-16}}{2} = -2 \pm 2i ]
This means our solutions are ( -2 + 2i ) and ( -2 - 2i ).
These complex numbers show us that there are still solutions for quadratic equations, even when they seem tricky!
When we solve quadratic equations, we sometimes find solutions that are complex. This happens when a part called the discriminant is negative.
The discriminant is figured out using this formula:
[ D = b^2 - 4ac ]
Let’s look at an example to understand this better.
Example 1:
For the equation ( x^2 + 4x + 8 = 0 ), we calculate the discriminant like this:
[ D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16 ]
Here, ( D ) is less than zero, which means we have complex solutions.
Finding the Solutions:
We use the quadratic formula to find the solutions. The formula is:
[ x = \frac{-b \pm \sqrt{D}}{2a} ]
Plugging in our numbers gives us:
[ x = \frac{-4 \pm \sqrt{-16}}{2} = -2 \pm 2i ]
This means our solutions are ( -2 + 2i ) and ( -2 - 2i ).
These complex numbers show us that there are still solutions for quadratic equations, even when they seem tricky!