Let's look at our objectives for today. We'll start by reviewing imaginary numbers. We'll then introduce complex numbers, another form of imaginary numbers. We'll review the quadratic formula. And finally we'll do some examples solving quadratic equations with non-real solutions.
Let's start by reviewing imaginary numbers. The square root of a negative number is non-real because any real number squared will not be negative. We defined the square root of negative 1 as the imaginary unit i. Imaginary numbers often arise when solving quadratic equations.
Now let's talk about complex numbers. A complex number is a value in the form a plus bi where a and b are real numbers and i is the imaginary unit. Notice that in the standard form for writing a complex number in the form a plus bi, the real part a is written first and the imaginary part bi is written second. For example, in the complex number 5 plus 3i, 5 is the real part of the complex number and 3i is the imaginary part where 3 is the coefficient and i is the imaginary unit. Complex numbers can occur when solving a quadratic equation using the quadratic formula.
Now let's review the quadratic formula. When solving a quadratic equation set equal to zero, the solution, or solutions, x, to the quadratic equation can be found using the quadratic formula. The variables a, b, and c in the formula correspond to the coefficients in the quadratic equation.
If the expression under the square root is negative, the quadratic equation will have zero real solutions. And when there are no real solutions to a quadratic equation, the graph of the equation will have 0 x-intercepts meaning the parabola will never intersect the x-axis.
The imaginary unit i is used to write the solutions of the quadratic equation as complex numbers.
Now let's do some examples solving quadratic equations with non-real solutions. We want to solve the quadratic equation x squared plus 16 equals 0. We start by subtracting 16 on both sides, which gives us x squared equals negative 16. We then cancel out the 2 exponent by taking the square root on both sides, which gives us x on the left side and positive and negative square root of negative 16 on the right side.
Remember that you need to include the positive and negative solutions when taking the square root. We then can use the product property of radicals to write the square root of negative 16 as the square root of 16 times the square root of negative 1. The square root of 16 is 4, and the square root of negative 1 is i. So our solutions are x equals positive 4i and negative 4i.
Let's do another example. We want to solve the quadratic equation x squared plus 5x plus 8 equals 0. We can solve this using the quadratic formula. To use the formula we identify our values for a, b, and c. X squared has no written coefficients so we know that it has an implied coefficient of 1, which means a equals 1. We see that b is 5, and c is 8. Substituting these values into our equation gives us the expression below.
We can start by simplifying our denominator 2 times 1 gives us 2. We then simplify our numerator starting with the expression underneath the square root. 5 squared minus 4 times 1 times 8 is equal to 25 minus 32. 25 minus 32 equals negative 7. This gives us x equals negative 5 plus or minus the square root of negative 7 over 2.
Because the expression underneath the square root is negative, our solution will be non-real. We can use the products property of radicals to write the square root of negative 7 as the square root of negative 1 times the square root of 7. This gives us i times the square root of 7. So we now have x equals negative 5 plus or minus i times the square root of 7 over 2.
We can separate our solution into two parts. We have negative 5 over 2 plus or minus i times the square root of 7 over 2. We cannot simplify our fractions any further because the numerator and denominator do not have any common factors other than 1. So this is our final solution.
Notice that our solution is a complex number in the form a plus bi, with the real part is negative 5/2, and the imaginary part is plus or minus i times square root of 7 over 2.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. The square root of a negative number is non-real because any real numbers squared will not be negative. When solving a quadratic equation, if the expression underneath a square root is negative, the quadratic equation has zero real solutions. Where there are zero real solutions to a quadratic equation, the graph of the equation will have zero x-intercepts, meaning the parabola will never intersect the x-axis. And finally, a complex number is a value in the form a plus bi where a is the real part, and b times i is the imaginary part of the complex number.
So I hope that these important points and examples helped you understand a little more about solving quadratic equations with non-real solutions. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.
00:00 – 00:35 Introduction
00:36 – 00:58 Imaginary Numbers
00:59 – 01:48 Complex Numbers
01:49 – 02:41 Quadratic Formula
02:42 – 06:08 Examples Solving Quadratic Equations with Non-Real Solutions
06:09 – 07:16 Important to Remember (Recap)
x = [-b ± sqrt(b^2-4ac)]/2a
sqrt(-1) = i