IIT JEE , AIEEE Forum - Mathematics , Algebra

rakesh61

for what values of a are the roots of the equation (a+1)x^2 - 3ax + 4a =0

a

-1 greater the unity

Please give me a detailed solution rating sure

dhwanitmunshi

the eqn is (a+1)x^2 - 3ax + 4a =0

now the roots of the equation should be > 1
i.e. X = -b +

( b^2 -4ac) > 1 (read + as + )
2a

3a +

(-7a²- 16a) > 1 (read + as + ) ( D =

b^2 -4ac Comes out as
2(a+1)

(-7a²- 16a )

3a +

(-7a²- 16a) > 2a + 2

3a +

-(7a²- 16a) > 2a + 2 OR 3a -

(-7a²- 16a) > 2a + 2

3a - 2a - 2 > -

(-7a²- 16a) OR 3a - 2a -2 >

(-7a²- 16a)

a - 2 > -

(-7a²- 16a) OR a - 2 >

(-7a²- 16a)
Squaring both sides

a^2 - 4a + 4 > -7a^2 - 16a OR a^2 - 4a + 4 > -7a^2 - 16a

a^2 + 7a^2 - 4a + 16a + 4 >0

8a^2 + 12a + 4 > 0

2a^2 + 3a + 1 >0

2a^2 + 2a + a + 1 > 0

2a(a+1) + 1(a+1) > 0

(2a+1)(a+1) > 0

2a+1 > 0 OR a+1 > 0

a > -1/2 OR a > -1

For a > -1/2 the roots of the Quadratic equation would be greater than unity

dhwanitmunshi

cmon pls have a look into this

lazycol

first check 4 D >= 0

ie 9a² - 4*4a*(a+1) >= 0 ie 7a² + 16a <= 0 so a(7a+16) <= 0

a [-16/7 , 0]..........................................................................(i)

let f(x) = (a+1)x² -3ax + 4a

CASE1: a+1 > 0 ie a>-1.......................................................(1)

f(1) shud b > 0; min value of f shud b at x > 1

f(1) = a+1-3a+4a = 2a+1 > 0 ie a>-1/2.............................(2)

min value at x = 3a/2(a+1) > 1 ie 3a>2(a+1) ie a>2.......(3)

combining (1),(2),(3)&(i) a has no sol

CASE2: a+1 < 0 ie a<-1.......................................................(1)

f(1) shud b < 0; max value of f shud b at x > 1

f(1) = a+1-3a+4a = 2a+1 < 0 ie a<-1/2.............................(2)

max value at x = 3a/2(a+1) > 1 ie 3a<2(a+1) ie a<2......(3)

combining (1),(2),(3)&(i) -16/7 <= a <-1

finally a [-16/7 , -1)

catch_arnnie

in short, 3 conditions are to be satisfied for the roots to be >1

1) D>0
2) -b/2a > 1
3) a * f(1) > 0

f(x)=(a+1)x^2 - 3ax + 4a =0

using the equation satisfy the above 3 conditions & you'll get the ans.

jaysunantony

roots greater than unity means (a+1)* f[x] > 0 and discriminant>=0

Solve these............. its playing with the graphs....................

jaysunantony

roots greater than unity means (a+1)* f[x] > 0 and discriminant>=0

Solve these............. its playing with the graphs....................

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