Question

$\sf If \: the \: equation \:4x^3+20x^2+23x+6=0$
$\sf \: have \: two \: of \: the \: roots \: \alpha \: \sf and \: \beta \:equal\: ,then$
$\sf \: \boxed{\sf\red{\gamma + { |\frac{a}{b}| }}}$ $\sf \: will \: be -$

$\bf Note- Consider,\: \gamma\: as \:third \: root$​

Answer 1

Answer:

4x

3

+20x

2

−23x+6=0

Let the roots be a,a,b

∑x=a+a+b=− 420 =−5

2a+b=−5

⇒b=−5−2a....(i)

∑xy=a(a)+a(b)+b(a)= 4−23 a 2+2ab= 4−23

substituting b from (i)

4a 2 +8ab+23=0

4a 2 +8a(−5−2a)+23=0

4a 2 −40a−16a 2 +23=0

12a 2 +40a−23=0

12a 2 −6a+46a−23=0

⇒a= 21 ,− 623

substituting a in (i)

⇒b=−6,− 38

∑xyz=a2 b=− 46=− 23

Product of roots do not satisfy a=− 38

So the roots of the equation are

21 , 21,−6