IM Alg2.3.18 Lesson: The Quadratic Formula and Complex Solutions
Mentally decide whether the solutions to each equation are real numbers or not.
Kiran was using the quadratic formula to solve the equation . He wrote this: Then he noticed that the number inside the square root is negative and said, “This equation doesn’t have any solutions.” Do you agree with Kiran? Explain your reasoning.
Write as an imaginary number.
Solve the equation 3x²-10x+50=0 and plot the solutions in the complex plane.
Although imaginary numbers let us describe complex solutions to quadratic equations, they were actually discovered and accepted because they could help us find real solutions to equations with polynomials of degree 3. In the 16th century, mathematicians discovered a cubic formula for solving equations of degree 3, but to use it they sometimes had to work with complex numbers. Let’s see an example where this comes up. To find a solution to the equation the cubic formula would first tell us to find a complex number, , which is . Find when our equation is .
The next step is to find a complex number so that . Show that works for the we found in step 1.
If we write i where and are real numbers, the solutions to our equation are , , and . What are the three solutions to our equation ?