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If a (x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct.True/False pls. help

 If a
(x – a) (x – c) + 2(x – b) (x – d) = 0 are real and distinct.True/False pls. help
 

Grade:upto college level

1 Answers

Navjyot Kalra
askIITians Faculty 654 Points
9 years ago
The given equation is
(x – a) (x – c) + 2(x – b) (x – d) = 0
⇒ 3x2 – (a + c + 2b + 2d)x + (ac + 2bd) = 0
For nature of roots, we find discriminant
D = (a + c + 2b + 2d)2 – 12(ac + 2bd)
= [(a + 2d)] – (c + 2b)]2 + 4(a + 2d) (c + 2b) – 12(ac + 2bd)
= [(a + 2d) – (c + 2b)]2 + 8ab + 8cd – 8ac – 8bd
= [(a + 2d) – (c + 2b)]2 + 8(c – b) (d – a) > 0. As given that
a < b < c < d ⇒ Statement is TRUE
ALTERNATE SOLUTION :
f(x) = (x –a) (x – c) + 2(x – b) (x – d).
f(a) = + ve
f(b) = - ve
f(c) = - ve
f(d) = + ve
∴ There exists two real and distinct roots one in the interval (a, b) and other in (c, d). Hence, (True).

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