As long as, x is not zero, we can transform algebraic form of x = 1/x into x^2 = 1 by multiplying both sides by x. This made a confusion in my campus. Some of my friends claim that x = 1/x is a quadratic equation. But, me and some other friends tend to say that this is not a quadratic equation.
Is there any reference that I can use to clarify this?
The equation x=1/x is an equation that can be reduced into a quadratic equation. It is given in any textbook that includes a chapter on quadratic equations.
To give a slightly more subtle interpretation: the two equations x^2=1 and x=1/x have precisely the same solutions. Any x for which one of the equations is true, the other is also true. In this instance, it is probably best to say that x=1/x is equivalent to the quadratic equation. However, there are other instances in which equations appear to be transformable to quadratic equations, but if you don't take care you miss special other solutions. Then the equations are different in a more fundamental way.
If the definition of a quadratic equation is ax^2+bx+c=0, then x=1/x is not a quadratic one.
Some of my friends are laughing when I posed this question to them. It seems that this is a silly question, a not important question. They are mathematician. As a mathematics educator, who see the existence of this misunderstanding phenomena, I try to know for sure what it is and why this is happening. My experience told me that a quadratic equation is an equation in form of ax^2 + bx + c = 0, where a is not zero. So, x = 1/x is not in the form of ax^2 + bx + c = 0. But, my friends keep saying that since this can be transformed into x^1 -1 = 0, then x = 1/x is a quadratic equation.
For me... the problem seems come from the words "in form of ax^2 + bx + c = 0" or "can be transferred into the form of ax^2 + bx + c = 0". These different perspective, of course, will lead to different understanding.
Why this is happening?
"Is a quadratic equation" and "Can be reduced to a quadratic equation" are two different things. If we have to solve the equation (x^2-x-1)(sin x+2)=0,we will "reduce" it to the quadratic equation x^2-x-1=0, but few people would dare to call the original equation quadratic.
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