Question

The quadratic equation abx²+acx+b(bx+c)=0 has non-zero equal and rational roots.The values of a and c respectively cannot be equal to what? (ab≠0)

Answer 1

It is given that the quadratic equation ab x²+ac x+b(bx+c)=0 has non-zero equal and rational roots.

→ ab x²+ac x+b(bx+c)=0

→ ax ( b x + c ) + b ( b x + c )= 0

→ ( a x  + b) ( b x + c) = 0

→ Either, ( a x  + b) = 0      ∨      ( b x + c) = 0

→ a x = 0- b                 ∨       b x = 0 - c

→ a x = - b    ∨     b x  = - c

x = -b/a   ∨   x= - c/b

The two values of x are -b/a and -c/b.

So, if a= 0, then x=-b/a is meaningless, and similarly if b=0, x=-c/b will become meaningless.→c can take any rational value.

But it is given that roots are equal.

→ -b/a = -c/b

→ (-b)× b = (-c)× a

→ - b² = - ac

Cancelling negative sign from both sides, we get

→ b² = ac