Question

17. For what value of m, the equation x^2 -mx + m + 1 = 0 may have its roots in the ratio 2 : 3​

Answer 1

Answer:-

m = 5 (or) -5/6

Expln:-

Roots in ratio 2:3

Then roots are 2a, 3a

Quadratic polynomial is in the form of,

f(x) = x²-(a+b)x+ab = 0 ..(1)

Given equ, x²-mx+(m+1) = 0

Frm (1),

a+b = m

2a+3a = m

5a = m ..(2)

ab = m+1

(2a)(3a)-1 = m

6a²-1 = m ..(3)

equ (2) and (3) are equal,

6a²-1 = 5a

6a²-5a-1 = 0

6a²-6a+a-1 = 0

6a(a-1)+1(a-1) = 0

(a-1)(6a+1) = 0

a = 1 (or) -1/6. in equ (2),

a = 1, 5a = m

m = 5

a = -1/6,

5(-1/6) = m

m = -5/6