Math Help - Rational Prob.

1. Rational Prob.

If (3+x)(3-x) is rational, then x is rational.

how would I claim that (3+x)(3-x) = p/q

Would I put (3+x)(3-x)/1=p/q -> q(3+x)(3-x)/p?

2. Re: Rational Prob.

Originally Posted by xmathlover
If (3+x)(3-x) is rational, then x is rational.

how would I claim that (3+x)(3-x) = p/q

Would I put (3+x)(3-x)/1=p/q -> q(3+x)(3-x)/p?
I don't think this is true, take $(3-x)(3+x)=1$ for instance. We have $x^2=8 \implies x =2\sqrt{2}$

The converse, however, is true. If $x=\frac{p}{q}$ then $(3+x)(3-x)=9-x^2=9-\frac{p^2}{q^2}=\frac{9q^2-p^2}{q^2}$

3. Re: Rational Prob.

I think you meant to write (3+x)/(3-x). Then this is true.

Assume that (3 + x)/(3 - x) is rational. Then (3 + x) / (3 - x) = p/q for some integers p and q where q =/= 0. Then cross multiply.

Thus, q(3 + x) = p(3 - x).
Then 3q + qx = 3p - px
Then px + qx = 3p - 3q
Then x(p + q) = 3(p - q)
Then x = 3(p - q)/(p + q)

Since p and q are integers, p - q and p + q are integers. Then 3(p - q) and p + q are integers. Then 3(p - q)/(p + q) is rational. Thus, x is rational.