Question

Sandra exclaims that her quadratic with a discriminant of -4 has no real solutions. Sandra then puts down her pencil and refuses to do any more work. Create an equation with a negative discriminant. Then explain to Sandra, in calm and complete sentences, how to find the solutions, even though they are not real. (10 points)

Answer 1

A simple example of this kind of quadratic is:\[\Large x^2+x+1=0\]Begin by plugging into the quadratic formula.\[\Large x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]a is 1, b is 1, and c is 1.\[\Large x=\frac{-1 \pm \sqrt{1^2-4(1)(1)}}{2(1)}\]\[\Large x=\frac{-1 \pm \sqrt{-3}}{2}\]It seems impossible to take the square root of -3, but split it up into -1 and 3 first.\[\Large x=\frac{-1 \pm \sqrt{-1}\sqrt{3}}{2}\]Call the square root of -1 i.\[\Large x=\frac{-1 \pm i \sqrt{3}}{2}\]So the solutions are:\[\Large x=\frac{-1 + i \sqrt{3}}{2}, x=\frac{-1-i \sqrt{3}}{2}\]

Answer 2

I realize that the original poster left, but I finished my response anyway, lol

Answer 3

1) Apologize for all of mathematics that some things just have stupid names. "Imaginary" is one of the worst. It doesn't mean they don't exist. Someone just gave them a confusing name. Don't let it bother you.

2) Explain that, one day, someone was wondering why such a simple equation as \(x^{2} + 1 = 0\) had no solution. This bugged someone so much that it was decide to invent a solution. It was called "i" for no particular reason. Maybe it was because "I" thought of it. In any case, \(x^{2} + 1 = 0\) now had a solution and a new world was born.

3) In modern parlance, we might express this same idea as \(\sqrt{-1} = i\) and call it a definition.

4) Reassure Miss Sandra that there will probably be other things in the future that won't seem quite right. If she gets caught in one way of thinking, trying to memorize processes, rather than understanding concepts, it will happen more and more.

5) Show how the Quadratic Formula is created via Completing the Square on an arbitrary Quadratic Equation.

Go!!!

Answer 4

I don't really understand what your saying, tkhunny.

Answer 5

Since you didn't say anything about my response, I'm assuming you didn't understand what I said either?

Answer 6

I kind of did but it doesn't say really what to say to Sandra!

Answer 7

What is there to understand? The question was to come up with something to say to Sandra. I gave you a whole script. Is there something about one of the five points that you find confusing? It's just a discussion. We're encouraging Sandra. It's NOT a calculation problem.