What is the square root of ax^2+bx+c?

sqrt(ax^2+bx+c)=? i tried myself and got a^(2)x+sqrt(c),a^(2)x+b/2 But i do not think That is exactly right.

1 Answer
Geoff K.
Feb 2, 2018

sqrt(ax^2+bx+c)=sqrt a" "x +sqrt c, as long as a and c aren't negative, and b=+-2sqrt(ac).

Explanation:

If ax^2+bx+c is a perfect square, then its square root is px+q for some p and q (in terms of a, b, c).

ax^2+bx+c = (px+q)^2
color(white)(ax^2+bx+c) =p^2" "x^2 + 2pq" "x + q^2

So, if we are given a, b, and c, we need p and q so that

p^2=a,
2pq = b, and
q^2=c.

Thus,

p=+-sqrt a,
q=+-sqrt c, and
2pq = b.

But wait, since p= +-sqrta and q=+-sqrtc, it must be that 2pq is equal to +-2sqrt(ac) as well, so ax^2+bx+c will only be a perfect square when b=+-2sqrt(ac). (Also, in order to have a square root, a and c must both be ge 0.)

So,

sqrt(ax^2+bx+c)=px+q
color(white)(sqrt(ax^2+bx+c))=sqrt a" "x +sqrt c,

if

a>=0,
c>=0, and
b=+-2sqrt(ac).

Related questions
Impact of this question
4966 views around the world
You can reuse this answer
Creative Commons License