# Without actually determining the roots comment upon the nature of the roots of each of the following equations:

1 view

Without actually determining the roots comment upon the nature of the roots of each of the following equations:

answered Feb 9 by (-4,325 points)

(i) 3x2 + 2x – 1 = 0

Here, a = 3, b = 2 and c = - 1

D = b2 – 4ac = 4 – 4 × 3 × (- 1)

⇒ D = 4 + 12 = 16 > 0

The given equation has real roots.

(ii) 2√3x2 - 2√2x - √3 = 0

Here, a = 2√3, b = - 2√2 and c = -√3

D = b2 – 4ac

⇒ D = 8 – 4 × 2√3 × - √3

⇒ D = 8 + 24 = 32 > 0

The given equation has real roots.

(iii) 9a2b2x2 – 48abcdx + 64c2d2 = 0

Here,   D = b2 – 4ac

⇒ ( - 48abcd)2 – 4 × 9a2b2 × 64c2d2

2304a2b2c2d2 – 2304a2b2c2d2 = 0

D = 0

Roots are real and equal.

(iv) x2 – 5x + 7 = 0

Here, a = 1, b = - 5 and c = 7

D = b2 – 4ac ⇒ 25 – 4 × 1 × 7

⇒ 25 – 28 = - 3

Since, D < 0 roots are imaginary.

(v) x2 – 4x + 1 = 0

Here, a = 1, b = - 4 and c = 1

D = b2 – 4ac   ⇒ 16 – 4 × 1 × 1

⇒ 16 – 4 = 12 > 0

The given equation has real roots.

(vi) x2 + 5x + 15 = 0

Here, a = 1, b = 5 and c = 15

D = b2 – 4ac = (5)2 – 4 × 1 × 15

= 25 – 60 = - 35  ⇒  D < 0  roots are imaginary.