b2 - 4ac = 0 can someone briefly explain thanks

This buisness about equal roots etc, when in a C1 question you need to show a proof by uising b2 - 4ac = 0 or greater, or not greater, what does that mean? so for example, when does it show it has equal roots, what does it mean when its smaller than zero etc?

thanks.

Reply 1

SsEe

13

b²-4ac < 0. Look at the formula. You can't take the square root of a negative number so this means there are no real roots (and the quadratic doesn't cross the x axis).

b²-4ac = 0. The roots are equal. The curve just touches the x axis at 1 point.

b²-4ac > 0. You can take the + or - square root so there are 2 real roots.

Reply 2

dvs

10

Are you familiar with the quadratic formula?

The roots of the equation ax²+bx+c are given by:
x = {-b ± sqrt[b²-4ac]}/2a

If b²-4ac=0, then x=-b/2a, i.e. one root.
If b²-4ac>0, then x=(-b ± k)/2a, i.e. two roots.
If b²-4ac<0, you'd take the square root of a negative number, so no real roots exist.

Reply 3

RichE

15

LondonBoy

This buisness about equal roots etc, when in a C1 question you need to show a proof by uising b2 - 4ac = 0 or greater, or not greater, what does that mean? so for example, when does it show it has equal roots, what does it mean when its smaller than zero etc?

thanks.

If you think about the quadratic formula for the solutions of

ax^2 + bx+c=0

you'll see that the roots are

(i) repeated/equal if b^2 - 4ac =0
(ii) distinct if b^2 - 4ac >0
(iii) complex and conjugate (or there are no real roots) if b^2 - 4ac <0, because here is a negative number you can't take the square root of (unless you have knowledge of complex numbers).

Reply 4

yazan_l

5

LondonBoy

This buisness about equal roots etc, when in a C1 question you need to show a proof by uising b2 - 4ac = 0 or greater, or not greater, what does that mean? so for example, when does it show it has equal roots, what does it mean when its smaller than zero etc?

thanks.

the equation has one root what it equals to zero coz :
x = [-b +- sqrt(b^2 - 4 a c ) ] / 2a
the sqrt of (b^2 -4ac) might be positive or negative if it was more than 0 , e.g. sqrt(9) = +/- 3
sqrt (4) = +/- 2
but sqrt (0) = 0
so there is only 1 value of x
when b^2 - 4 a c is less tahn 0, then it has no real sqrt, so there is no real value for x

Reply 5

dvs

10

It's busy on here tonight. :smile:

Reply 6

yazan_l

5

LondonBoy

This buisness about equal roots etc, when in a C1 question you need to show a proof by uising b2 - 4ac = 0 or greater, or not greater, what does that mean? so for example, when does it show it has equal roots, what does it mean when its smaller than zero etc?

thanks.