(i) 3x^{2} + 2x – 1 = 0
Here, a = 3, b = 2 and c = - 1
D = b^{2} – 4ac = 4 – 4 × 3 × (- 1)
⇒ D = 4 + 12 = 16 > 0
The given equation has real roots.
(ii) 2√3x^{2} - 2√2x - √3 = 0
Here, a = 2√3, b = - 2√2 and c = -√3
D = b^{2 }– 4ac
⇒ D = 8 – 4 × 2√3 × - √3
⇒ D = 8 + 24 = 32 > 0
The given equation has real roots.
(iii) 9a^{2}b^{2}x^{2} – 48abcdx + 64c^{2}d^{2} = 0
Here, D = b^{2} – 4ac
⇒ ( - 48abcd)^{2} – 4 × 9a^{2}b^{2} × 64c^{2}d^{2}
2304a^{2}b^{2}c^{2}d^{2} – 2304a^{2}b^{2}c^{2}d^{2} = 0
D = 0
Roots are real and equal.
(iv) x^{2} – 5x + 7 = 0
Here, a = 1, b = - 5 and c = 7
D = b^{2} – 4ac ⇒ 25 – 4 × 1 × 7
⇒ 25 – 28 = - 3
Since, D < 0 roots are imaginary.
(v) x^{2} – 4x + 1 = 0
Here, a = 1, b = - 4 and c = 1
D = b^{2 }– 4ac ⇒ 16 – 4 × 1 × 1
⇒ 16 – 4 = 12 > 0
The given equation has real roots.
(vi) x^{2} + 5x + 15 = 0
Here, a = 1, b = 5 and c = 15
D = b^{2} – 4ac = (5)^{2} – 4 × 1 × 15
= 25 – 60 = - 35 ⇒ D < 0 roots are imaginary.