Quod Erat Demonstrandum

2011/09/19

(x=2 or x=-2) implies x^2=-4?

Filed under: Junior Form Mathematics,NSS — johnmayhk @ 4:11 下午

Just a question asked by a student, Thomas.

x=2
x=-2

multiply the above, yield

x^2=-4

strange?

He puzzled that, why we need to do something like:

x-2=0
x+2=0

to obtain

(x-2)(x+2)=0
\Rightarrow x^2=4

instead of considering the product of

x=2 and x=-2 directly?

The above procedure may be found under the context of forming equation, like

“Determine a quadratic equation with roots 2 and -2."

The crucial idea is, for real numbers a and b,

(a=0 or b=0) if and only if ab=0.

Note that, ZERO plays a crucial role to maintain the equivalence of the two statements (a=0 or b=0) and ab=0.

Let me make it clear, as for example,

the statement (a=2 or b=3), DOES NOT imply ab=6.

(a=2 or b=3) is true even if (a=2 and b=5) (say), hence, in this case, ab=(2)(5)=10 (not 6).

Further, the statement (a=2 and b=3) implies ab=6 indeed. However, ab=6 DOES NOT imply (a=2 or b=3), because, ab=6 may imply other cases like (a=6 and b=1), (a=12 and b=0.5), and so on.

We see that

(a=2 or b=3) DOES NOT imply ab=6.

or we say,

(a=2 or b=3) is not equivalent to ab=6.

Back to the original question,

(x=2 or x=-2) is not equivalent to x^2=-4.

however,

(x=2 or x=-2) is equivalent to (x-2=0 or x+2=0) and hence equivalent to (x-2)(x+2)=0 \Leftrightarrow x^2=4.

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