Now we will begin to explore and solve quadratics with complex solutions. Complex solutions occur when the square root of a negative number is required. We have previously stated this has no real solution, but it actually has a complex solution or imaginary solution.
Before we begin solving quadratics with complex solutions we need to learn and practice some skills with imaginary numbers.
Before we begin solving quadratics with complex solutions we need to learn and practice some skills with imaginary numbers.
Imaginary Numbers: Any time we multiply a number by itself, the product will always be positive or zero. The product will never be negative. So how can we take the square root of a negative number? This is where the imaginary number comes in.
By definition, i² = -1, therefore i = √-1.
By definition, i² = -1, therefore i = √-1.
Simplifying Roots of Negative Numbers: Now we know that we can take the square root of a negative number using i, lets do some practice. Notice in the examples below, the only difference is that an i comes out when you see a negative under the radical. Copy the examples into your notes then do the Khan Academy practice. Help videos are included in the practice.
Powers of i: When working with imaginary numbers, often times you will come across higher powers of i.
Using either of the methods from above, the same concept will be used. Since any power of i can be simplified to 1, i , -1, or -i, any power that is divisible by 4 is = 1. Simply dividing the power by 4 and finding the remainder will give us what we are looking for.
Complex numbers: Complex numbers have a real and imaginary part. The link below is a lesson on complex numbers. While viewing the lesson make sure to take notes and complete the practice problems that are mixed in.
Operations with Complex numbers: Adding, subtracting, and multiplying complex numbers uses the same properties of adding, subtracting and multiplying polynomials. Copy the example below into your notes then complete the practice skills.