Exactly one root of the equation x^(2)- (a+3)x+4=0 lies between 3 and 4 for a in(m n) then m+n is ( where x denotes greatest integer less than or equal to x )
Answers
Given : Exactly one root of the equation x² - (a + 3)x + 4 = 0 lies between 3 and 4 for a ∈ (m , n)
To Find : m + n
Solution:
x² - (a + 3)x + 4 = 0
=> x = ( (a + 3) ± √(a + 3)² - 4(4) )/2
=> x = ( (a + 3) ± √a² + 6a - 7)/2
a² + 6a - 7 = (a + 7)(a - 1)
(a + 7)(a - 1) ≥ 0
=> a≥ 1 , a ≤ - 7
x² - (a + 3)x + 4 = 0
one root lies between 3 and 4
Product of roots is 4
Hence another root lies between 1 & 4/3
Sum of roots = a + 3
Range of Sum of roots = 4 + 1 = 5 or 3 + 4/3 = 13/3
13/3 < a + 3 < 5
=> 4/3 < a < 2
a ∈ ( 4/3 , 2 )
m = 4/3
n = 2
4/3 + 2 = 10/3
m + n = 10/3
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