Find all values of p , so that 6 lies between roots of the equation x2 + 2(p-3)x + 9 = 0
Dear Student,
x2 + 2(p-3)x + 9 = 0
Let m & n be two roots of the equation
Comparing given equation with ax2 + bx + c =0
We get, a = 1 , b = 2(p-3) , c = 9
Now,
a) D > 0
b2 - 4ac > 0
[2(p-3)]2 - 4(1)(9) > 0
4(p2 - 6p + 9) - 36 > 0
4p2 - 24p + 36 - 36 > 0
p2 - 6p > 0
p (p - 6) > 0
p (- , 0) (6 , ) ....................... (1)
b) a f(6) < 0
1 . [62 + 2(p-3)6 + 9] < 0
36 + 12p - 36 + 9 < 0
4p + 3 < 0
p < -
p ( - , - ) .................................. (2)
c) Let m and n be the two roots of the equation
sum of roots = -2(p-3)
As per given condition: m < 6 ...... (1)
n > 6⇒ -n < -6 ......... (2)
Subtracting (2) from (1) we get,
m + n < 12
-2(p-3) < 12
p-3 > 6
p > 9
p (9 , ) ........................... (3)
hence values of p satisfying (1) , (2) and (3) is p ( - , - ) (9 , )
Regards
x2 + 2(p-3)x + 9 = 0
Let m & n be two roots of the equation
Comparing given equation with ax2 + bx + c =0
We get, a = 1 , b = 2(p-3) , c = 9
Now,
a) D > 0
b2 - 4ac > 0
[2(p-3)]2 - 4(1)(9) > 0
4(p2 - 6p + 9) - 36 > 0
4p2 - 24p + 36 - 36 > 0
p2 - 6p > 0
p (p - 6) > 0
p (- , 0) (6 , ) ....................... (1)
b) a f(6) < 0
1 . [62 + 2(p-3)6 + 9] < 0
36 + 12p - 36 + 9 < 0
4p + 3 < 0
p < -
p ( - , - ) .................................. (2)
c) Let m and n be the two roots of the equation
sum of roots = -2(p-3)
As per given condition: m < 6 ...... (1)
n > 6
Subtracting (2) from (1) we get,
m + n < 12
-2(p-3) < 12
p-3 > 6
p > 9
p (9 , ) ........................... (3)
hence values of p satisfying (1) , (2) and (3) is p ( - , - ) (9 , )
Regards