"How can I prove that if x2+bx+c is..." was answered by our community

jncuenodd4nf

Answered question

2022-03-27

How can I prove that if

x^{2} + b x + c

is factorable, then

x^{2} - b x + c

is also factorable?

Jesse Gates

Beginner2022-03-28Added 19 answers

Step 1
If
1)

x^{2} + b x + c = (x - p) (x - q)

then since
2)

(x - p) (x - q) = x^{2} - (p + q) x + p q

We find that
3)

b = - (p + q)

c = p q

now reverse the signs of p and q, and compute
5)

(x + p) (x + q) = x^{2} + (p + q) x + p q = x^{2} - b x + c

which shows that

x^{2} - b x + c

is factorable provided

x^{2} + b x + c

is
Note that, replacing b with -b, we see that

x^{2} + b x + c

if factorable provided

x^{2} - b x + c

is
It is worth observing, I think, that this result holds for
5)

x^{2} + b x + c \in R [x],

where R is any commutative ring; indeed, it may be possible to extend the coefficient ring even further; certainly something to ponder.

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?