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```        let a, b and c are unequal real positive numbers such that 2b = a + c, then roots of ax^2 + 2bx + c = 0 are
(i) real and equal (ii) Real and distinct (iii) imaginary (iv) nothing definite can be said
Plz explain```
8 years ago

## Answers : (3)

```							Dear student,
2b=a+c
Sum of the roots=-2b/a
Product=c/a
This shows the roots are real and distinct...

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```
8 years ago
```							for any quadratic eq  ax2+bx+c roots are given by
X = {-b+_ D)/2a
D =discriminant = (b2-4ac)1/2
our eq is ax2+2bx+c so
D = {(2b)2-4ac)1/2 ........1
2b = a+c  (given)
so , (2b)2 = (a+c)2 ...........2
putting  eq2 in eq1
D = { (a+c)2 - 4ac}1/2  =  { a2+c2-2ac}1/2 ={(a-c)2}1/2 =a-c
now roots are
X = (-2b +_ D}/2a = {-2b + (a-c)}/2a     or        {-2b-(a-c)}/2a
= -c/a                      or        -1                                  (on putting 2b =a+c )
so the roots are unequal & real
option (ii) is correct
```
8 years ago
```							the equation is ax^2+2bx+c=0
the discriminant is= B^2-4AC
A=a,B=2b,C=c
discriminant= 4b^2-4ac
but 2b=a+c, therefore (2b)^2=(a+c)^2
= (a+c)^2-4ac
=a^2+c^2+2ac-4ac
=a^2+c^2-2ac
=(a-c)^2, which is always greater than zero
therefore roots are real and distinct
```
8 years ago
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