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Let [a] denote the greatest integer less than or equal to a. given that the quadratic equation x2 + [a2 - 5a + b + 4]x +b = 0 has roots -5 and 1 , find the no. Of integral values a

Let [a] denote the greatest integer less than or equal to a. given that the quadratic equation x2 + [a2 - 5a + b + 4]x +b = 0 has roots -5 and 1 , find the no. Of integral values a

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4 Answers

ADARSH SINGH
123 Points
5 years ago
DEAR FRIEND ,
ACTUALLY , NO INTEGRAL VALUE OF ‘a’ EXISTS FOR GIVEN CONDITION.
SOLUTION:-
Since , 1 is solution of given equation, on substituting 1 in place of x , you should get 0.
hence you get equation:- 
[a^2-5a+ b] +b=-5
now the  values inside the brackets are integral and when b is added ,we get integral value -5. so , b must be integral. hence we can take out b from the greatest integer brackets as it is an integer? hence , we get following equation 
[a^2-5a]+2b=-5
or .2[a^2-5a]+4b=-10                 ............equation 1.
now -5 is also solution of given equation. substituting -5 in place of x and considering b as an integer and taking it out of greatest integer brackets we get equation-
5[a^2-5a]+4b=5                   ..........equation 2.
on subtracting equation 1 frrom equation 2, we get 
3[a^2-5a]=15
or [a^2-5a]=5
there fore , since greatest integer function on a^2-5a is 5,
5\leq a^2-5a< 6
on observation and hit and trial , we will come to conclusion that no integral value of a exists for above condition. hope you got your answer. 
FOR ANY QUERY , PLEASE ASK FREE.
THANK YOU.
Arun
25750 Points
5 years ago
Sum of roots = -4 = -[a² -5a + b +4]
And 
Product of roots = -5 = b
Now
Solve these two equations
[a² -5a -1] = 4
4
Hence
5
Now solve it and you will find 
Values of a as-
(-1, {(5 - 3 * sqrt(5)) /2}] U [{(5 + 3* srt(5)) /2}, 6)
ADARSH SINGH
123 Points
5 years ago
ARUN,
I DON’T THINK THAT YOU HAVE ANSWERED RIGHT , BECAUSE THE QUESTION IS THAT HOW MANY INTEGRAL VALUES OF A EXIST !
BUT THE RANGE OF A YOU HAVE PROVIDED IS 
(-1, {(5 - 3 * sqrt(5)) /2}] U [{(5 + 3* srt(5)) /2}, 6)
IN THIS RANGE NO INTEGRAL VALUE OF A EXISTS . SO ACTUALLY THE ANSWER IS THAT THERE WILL BE NO INTEGRAL VALUE OF A.
PLEASE CORRECT ME IF I AM WRONG .
Arun
25750 Points
5 years ago
Sorry
 
And thanks Adarsh for making me correct.
Actually I think value of a is asked. But integral value of a is asked.
Hence no integral value is obtained.

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