Question

Given that the roots of the quadratic equation ax^2+bx+c=0 are real, explain by inspection of the coefficients how one can determine:

(a) whether the roots have opposite signs
(b) the sign of the roots if they both have the same sign
(c) the sign of the numerically greater root if they have opposite signs.

Answer 1

want to work by example?

Answer 2

yea, sure :)

Answer 3

pick two roots of the same sign, lets make them both positive
say \(2\) and \(3\) to make it easy

Answer 4

the quadratic equation that has those two roots is
\[(x-2)(x-3)\] which, when multiplied out, is
\[x^2-5x+6\]

Answer 5

now lets pick to negative roots, say \(-2\) and \(-3\) and multiply
\[(x+2)(x+3)=x^2+5x+6\]
what do you notice?

Answer 6

it will be greater than 0

Answer 7

what is the "it" ?

Answer 8

alpha and beta

Answer 9

assuming for a moment that the leading coefficient is 1
then the product of the roots is the constant
so if the constant is positive, they must have the same sign, i.e. both be positive, or both be negative

Answer 10

so suppose you are looking at
\[x^2+4x+2\] where the constant is 2
both roots must therefore have the same sign
do you think they are positive or negative?

Answer 11

positive

Answer 12

hmmm
each coefficient is positive right?

Answer 13

yea

Answer 14

so if you plug in any positive number for \(x\) is it not pretty clear that you have to get a positive number out?

Answer 15

yea

Answer 16

ok so if you have to get a positive number out, it cannot be zero can it?

Answer 17

what i am trying to say is if all the coefficients are positive, and you have to zeros of the same sign, they must both be negative

Answer 18

that is why \(x^2+5x+6\) had two negative roots, wheres
\[x^2-5x+6\] had two positive roots

Answer 19

and if the constant is negative, like say if the zeros are \(2\) and \(-3\) so the quadratic is
\[(x-2)(x+3)=x^2+x-6\] then you can see that the roots have opposite sign

Answer 20

of course you need to check first whether your root are real or not. for example you can't say roots are negative because all coeff. are positive.\[
x^2+4x+5=0
\]
first check discriminant that is \[ \Delta = b^2-4ac\]

if \[ \Delta>0 \] roots are real and different.
if \[ \Delta=0 \] roots are real and they are same.
if \[ \Delta<0 \] roots are not real (i.e. they are complex).

Answer 21

\[
x^2+4x+5=0\\

\Delta=4^2-4*1*5=16-20=-4
\]

so roots are not real..

Answer 22

sorry it is already given real roots, but anyway..