Question

Solve :

Answer 1

solved

Answer 2

1.) The real value of "a" for which the sum of the squares of the roots of the eq. x^2-(a-2)x-a-1=0 assume the least value is ??
Ans. 1

2.) If log(base2)(ax^2+x+a)>=1 is true for all real x,then exhaustive set of values of "a" is-
Ans. [ 1+[root(5)]/2 , infinity)

Solution required of both questions.

Answer 3

@nitz @ganeshie8 @mukushla

Answer 4

can u plz clear the question 1 ?

Answer 5

ok i got it

Answer 6

i've written it as written in my book.
the eq. is \[x^{2}-(a-2)x-a-1=0\]
we have to find the least value of the squares of this eq.

Answer 7

*of the sum of the squares of roots of that eq.

Answer 8

assume that roots are \( \alpha \) and \( \beta \) so we have : \( \alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha\beta \)

on the other hand we have : \( \alpha+\beta=a-2 \) and \( \alpha \beta=-(a+1) \)
am i right?

Answer 9

then....

Answer 10

\( \alpha^2+\beta^2=(a-2)^2+2(a+1)=a^2-2a+6\)
can u compute the minimum of this quadratic function?

Answer 11

complete the squre
or
use the vertex formula x=-b/2a
or
take derivative

Answer 12

vertex formula??

Answer 13

http://www.algebra-class.com/vertex-formula.html

Answer 14

and y=?

Answer 15

u need to find the value of a for which a^2-2a+6 is minimum .......what is y?

Answer 16

i just want to know the complete vertex formula

Answer 17

i know it was something like -D/...

Answer 18

vertex formula of quadratic function
\[\huge y=ax^2+bx+c\]
is
\[\huge (x,y)=(-\frac{b}{2a} \ , \ -\frac{b^2-4ac}{4a} )\]

Answer 19

ok

Answer 20

now can you help me with log ques.

Answer 21

guys help :D

Answer 22

for the second one we must have \(ax^2+x+a \ge 2 \) or \(ax^2+x+a-2 \ge 0 \)

Answer 23

ok then

Answer 24

so the Discriminant of the last quadratic equation must be less than or equal to zero

Answer 25

which last eq.?

Answer 26

ax^2+x+a-2

Answer 27

do u know why?

Answer 28

ok then

Answer 29

yes i know

Answer 30

and also a>0 because if we want ax^2+x+a-2 to be greater or equal than zero the coefficient of x^2 must be greater than zero

Answer 31

yes then

Answer 32

now u just need to solve this to inequality simultaneously

a>0 and 1-4a(a-2) <=0

Answer 33

after solving both these , what sol. set you got?