Determine the nature of the roots of the following quadratic equations:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
, 
(viii) 
(ix) 
(i) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 9 – 4 × 5 × 2 = -31
Roots are not real.
(ii) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 36 – 4 × 2 × 3 = 12
Roots are real and distinct.
(iii) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 4/9 – 4 × 3/5 × 1 = -88/45
Roots are not real.
(iv) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 48 – 4 × 3 × 4 = 0
Roots are real and equal
(v) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 24 – 4 × 3 × 2 = 0
Roots are real and equal.
(vi) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ x2 – (2a + 2b)x + 4ab = 4ab
⇒ x2 – (2a + 2b)x = 0
D = (2a + 2b)2 – 0 = (2a + 2b)2
Roots are real and distinct
(vii) 
, 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 576a2b2c2d2 – 4 × 16 × 9 × a2b2c2d2 = 0
Roots are real and equal
(viii) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = 4(a + b)2 – 4 × 2 × (a2 + b2)
⇒ D = -4(a2 + b2) + 2ab = -(a – b)2 – 3(a2 + b2)
Roots are not real
(ix) 
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D < 0, roots are not real
If D > 0, roots are real and unequal
If D = 0, roots are real and equal
⇒ D = (a + b + c)2 – 4a(b + c)
⇒ D = a2 + b2 + c2 – 2ab – 2ac + 2bc
⇒ D = (a – b – c)2
Thus, roots are real and unequal