When factoring the quadratic \(\displaystyle{x}^{{{2}}}+{b}{x}+{c}\), where b

Alisa Duarte

Alisa Duarte

Answered question

2022-03-13

When factoring the quadratic x2+bx+c, where b and c are integers, why do we not consider the case where u and v in the factored form (x+u)(x+v) are fractions?

Answer & Explanation

bioadferatxh

bioadferatxh

Beginner2022-03-14Added 7 answers

Notice that if u=pq and v=mn for some integers p, q, m, n then you obtain the equation
qn(x+pq)(x+mn)=qnx2+(mq+pn)x+±.
In the expression qnx2+(mq+pn)x+± we can conclude that qn,mq+pn, and pm are all integers. If we use the labels
 

a = q n b = m q + p n c = p m u = p q v = m n


then we again end up with
(xu)(xv)=ax2+bx+c,
so in reality this form covers your case as well. Generally, though, we don't require a, b, c, u, v to be integers.

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