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

x1 = - b/a & x2 = - (ac+b^2) / ab

Answer 2

The given Quadratic equation is

    abx²+acx+b(bx+c)=0

⇒ a b x² + ac x+ b² x + b c=0  [Using distributive property]

⇒ a x(b x + c) + b( bx+c)=0

⇒ (a x + b)(b x+ c)=0

⇒ a x + b=0 ∧ b x+ c =0

⇒ a x = - b    ∧   b x =- c

  x = -b/a    ∧    x= -c/b

For ,the value of x to exist , i.e if the quadratic equation has non zero equal and rational roots, then

→-b/a = -c/b

Cancelling negative sign from both sides

→b×b=a×c

→ b² = ac

So , Solution which satisfies the above condition is b²=ac