**(i)** \((x-2a)(x-2b)=4ab\)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 4ac

If D < 0, roots are not real

If D > 0, roots are real and unequal

If D = 0, roots are real and equal

\((x-2a)(x-2b)=4ab\)

⇒ x^{2} – (2a + 2b)x + 4ab = 4ab

⇒ x^{2} – (2a + 2b)x = 0

D = (2a + 2b)^{2} – 0 = (2a + 2b)^{2}

**Roots are real and distinct**

**(ii) **\(9a^2b^2x^2-24abcdx+16c^2d^2=0,a\neq0,b\neq0\)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 4ac

If D < 0, roots are not real

If D > 0, roots are real and unequal

If D = 0, roots are real and equal

\(9a^2b^2x^2-24abcdx+16c^2d^2=0,a\neq0,b\neq0\)

⇒ D = 576a^{2}b^{2}c^{2}d^{2} – 4 × 16 × 9 × a^{2}b^{2}c^{2}d^{2} = 0

**Roots are real and equal**

**(iii)** \(2(a^2+b^2)x^2+2(a+b)x+1=0\)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 4ac

If D < 0, roots are not real

If D > 0, roots are real and unequal

If D = 0, roots are real and equal

\(2(a^2+b^2)x^2+2(a+b)x+1=0\)

⇒ D = 4(a + b)^{2} – 4 × 2 × (a^{2} + b^{2})

⇒ D = 4(a + b)^{2} – 4 × 2 × (a^{2} + b^{2})

**Roots are not real**

**(iv)** \((b+c)x^2-(a+b+c)x+a=0\)

For a quadratic equation, ax^{2} + bx + c = 0,

D = b^{2} – 4ac

If D < 0, roots are not real

If D > 0, roots are real and unequal

If D = 0, roots are real and equal

\((b+c)x^2-(a+b+c)x+a=0\)

⇒ D = (a + b + c)^{2} – 4a(b + c)

⇒ D = a^{2} + b^{2} + c^{2} – 2ab – 2ac + 2bc

⇒ D = (a – b – c)^{2}

**Thus, roots are real and unequal**