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What is the value of x+y ?

## Progressions and Equations

Let x3- x2 + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true
1. b=2,c=2
2. b=1,c=1
3. b= -1,c = 1
4. b= -1,c= -1

Choice B. 9

## Detailed Solution

Given the roots are in A.P. so let a-d, a, a+d be the roots
From equation, sum of roots = 1
Sum of two roots taken at a time = +b
Product of two roots = -c
∴ (a-d)+ (a)+ (a+d) = 1
=) 3a = 1
=) a = $\frac {1}{3}$
Also, (a-d)a+ a(a+d)+ (a-d)(a+d) = b
=) a2 – ad + a2 + ad + a2 – d2 = b
=) 3a2 – d2 = b
=) 3x$\frac {1}{9}$ – d2 = b
=) d2 = b - $\frac {1}{3}$
Now, since d is a real number,
$\frac {1}{3} - b > 0 =) b < \frac {1}{3}$
Also,
(a-d)(a)(a+d) = -c
=) a(a2 – d2) = -c
=) $\frac {1}{3} * (\frac {1}{9} - d^2) = -c$
=) $c = \frac {d^2}{3} - \frac {1}{27}$
=) $c = \frac {d^2 - 1}{27}$
=) $d^2 = 3c + \frac {1}{9}$
Again, $3c + \frac {1}{9} > 0$
$c > \frac {-1}{27} > 0$
∴ Only option (c) b= -1 , c =1 could be true .

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