Question

If a < b < c < d , then the roots of the equation (x – a) (x – c ) + 2 (x – b) (x – d) = 0 are

• A. Real and distinct
• B. Real and equal
• C. Imaginary
• D. None of these

Answer 1

The correct option is A

Real and distinct

Given equation can be rewritten as

$3x^2$ – (a + c + 2b + 2d)x + (ac + 2bd) = 0

Its discriminant D

=$(a+c+2b+2d{)}^2$ – 4.3(ac + 2bd)

= $\left\{\right(a+2d)+(c+2b){\}}^2$ – 12(ac + 2bd)

= $\left\{\right(a+2d)–(c+2b){\}}^2$ + 4(a + 2d)(c + 2b) – 12(ac + 2bd)

= $\left\{\right(a+2d)–(c+2b){\}}^2$ + 8(c – b)(d – a)

Which is +ve, since a < b < c < d. Hence roots are real and distinct.