Today we're going to talk about solving quadratic equations with non-real solutions. So we're going to start by looking at what it means to be a quadratic equation with no real solution, and then we'll do some examples solving quadratic equations.
So let's start by looking at what it means to have a non-real solution for a quadratic equation. I've got here the quadratic formula, which, if you remember, is one method of solving a quadratic equation. And in our formula, we can specifically look at the expression which is underneath the square root symbol. We call this the discriminant. And the discriminant will tell you whether your solution for the equation is real or not real.
If the value of the discriminant is equal to 0 or greater than 0, then you're going to have a real solution for your quadratic equation, or real solutions for your quadratic equation. If the value of the discriminant is less than 0, strictly less than 0, then you're going to have non-real solution for your quadratic equation.
So if we have a quadratic equation with a non-real solution, it doesn't mean that the quadratic equation doesn't have a solution. It just means that the solution are non-real numbers. We call non-real numbers imaginary numbers. And we can define the term "i" as the square root of negative 1. So i is defined as the square root of negative 1. And we say that i is imaginary because there's no real number squared, or there's no real number multiplied by itself, that will give you a value of negative 1. So the square root of negative 1 is imaginary. It's not a real number.
So we have ways of writing terms that are imaginary. So, for example, if we have the square root of negative 5, we know that this is going to be an imaginary number, and so I can rewrite the square root of negative 5 as the square root of 5 times the square root of negative 1, using my product property of radicals. And I can simplify the square root of negative 1 to just be i. So this expression becomes the square root of 5 i.
Another example would be the square root of negative 16. So again, this is an imaginary number. I can rewrite it as the square root of 16 times the square root of negative 1, and again s substituting i in for the square root of negative 1, I'll have the square root of 16 i. And the square root of 16 is just 4, so this expression becomes 4i.
So let's do an example, solving a quadratic equation that will have no real solution. So if I were to solve the quadratic equation "0 is equal to 2x squared minus 4x plus 4," I know first that if I were to graph the equation "y equals 2x squared minus 4x plus 4," the graph will look like this. And so we can see that because there are no x-intercepts, from the graph we already know that there are no real solutions.
Now let's see how we can verify that by using the quadratic formula to solve this equation. So in my quadratic equation, I need values for a, b, and c. So I see that my value for a will be 2, b will be negative 4, and c will be positive 4. So substituting those three values into my quadratic formula, I'll have negative b, so negative negative 4 will become positive 4, plus or minus the square root of b squared, so negative 4 squared, minus 4 times my value of a, so 2 times my value of c, so positive 4, and that's all over 2 times my value for a.
Now I'll simplify this first by simplifying underneath the square root. So I'll have 4 is-- oh, I'm sorry. 4 plus or minus the square root of negative 4 squared is positive 16, minus-- I can multiply these three numbers together. 4 times 2 is 8, times 4 will give me 32, so I'll have 16 minus 32. Over-- I'm going to go ahead and simplify the denominator too, so that will just give me 4.
Then continuing to simplify underneath my square root, I've got 16 minus 32, which will give me negative 16, and because I know that I can rewrite the square root of negative 16 as the square root of 16 times the square root of negative 1, and that the square root of 16 is 4, and the square root of negative 1 is i, I can rewrite this as 4i. So this will become 4 plus or minus 4i over 4.
Now I can simplify this because I have a common factor in all of my terms. So if I were to factor out a 4 from the terms in my numerator, I'll have 1 plus or minus 1i, or just i. And then these two 4's will cancel, leaving me with just 1 plus or minus i for my solution.
Now this solution is written as a complex number. Complex numbers have both a real number component and an imaginary number component. So 1 would be my real number component, and i is my imaginary number component. So in general, complex numbers can be written as "a plus or minus bi," where a and b could be any real numbers.
So let's go over our key points from today. In the quadratic formula, if the discriminant is greater than or equal to 0, then the solutions to the quadratic equation will be real numbers. If the discriminant is less than 0, the equation has no real solution. Looking at the graph of a quadratic equation, if the parabola does not cross or intersect the x-axis, then the equation has no real solution. And no real solution does not mean that there is no solution, but that the solutions are not real numbers.
So I hope that these key points and examples helped you understand a little bit more about solving quadratic equations with no real solution. Keep using your notes and keep on practicing, and soon you'll be a pro Thanks for watching.
a number containing a real component and an imaginary component
a non-real number which is a multiple of i, sqrt(-1)