A bit less than perfect

Relevant wiki: FOIL Method

We can disprove Alice, Ben, Charlie, and Emily through counterexamples:

"Since $x^{2}-1$ is a product of 2 numbers, all numbers 1 less than a perfect square are composite."

Set $x$ equal to 2: $2^{2}-1=4-1=3$ 3 is NOT composite. Therefore, the statement is false.

There is also a flaw in the statement to begin with, since composite numbers aren't just a product of two numbers. They are a product of at least two integers greater than 1.

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"All numbers 1 more than a perfect square are prime."

Set $x$ equal to 3: $3^{2}+1=9+1=10$ 10 is NOT prime. Therefore, the statement is false.

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"The lowest possible value for $x^{2}-1$ , where $x$ is any number, is $-1$ ."

Set $x$ equal to $\sqrt{-1}$ (aka the imaginary number, $i$ ): $i^{2}-1=-1-1=-2$ $\boxed{-2<-1}$

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"the equation itself is wrong."

Using the FOIL method, $(x+1)\times(x-1)=x^{2}+x-x-1$ $x^{2}+x-x-1=\boxed{x^{2}-1}$

$\beta_{\lceil \mid \rceil}$