Examine the nature of the roots of the equations: (i) 2x2 + 2x + 3 =

1.	Examine the nature of the roots of the equations: (i) 2x2 + 2x + 3 = 0 (ii) 2x2 – 7x + 3 = 0 (iii) x2 – 5x – 2 = 0 (iv) 4x2 – 4x + 1 = 0
Answer» (i) 2x² + 2x + 3 = 0 (Here, a = 2, b = 2, c = 3) ∴ D = b² – 4ac = (2)² – 4 × 2 × 3 = 4 – 24 = – 20 < 0 Hence, roots are imaginary. (ii) 2x² – 7x + 3 = 0 (Here, a = 2, b = – 7, c = 3) ∴ D = b² – 4ac = 49 – 24 = 25 > 0 and a perfect square Hence, roots are real and rational. (iii) x² – 5x – 2 = 0. (Here, a = 1, b = – 5, c = – 2) ∴ D = b² – 4ac = 25 + 8 = 33 > 0 and not a perfect square Hence, roots are real and irrational. (iv) 4x² – 4x + 1 = 0 (Here, a = 4, b = – 4, c = 1) ∴ D = b² – 4ac = 16 – 16 = 0. Hence, roots are real and equal.

Examine the nature of the roots of the equations: (i) 2x2 + 2x + 3 = 0 (ii) 2x2 – 7x + 3 = 0 (iii) x2 – 5x – 2 = 0 (iv) 4x2 – 4x + 1 = 0

Answer»

(i) 2x² + 2x + 3 = 0 (Here, a = 2, b = 2, c = 3)

∴ D = b² – 4ac = (2)² – 4 × 2 × 3 = 4 – 24 = – 20 < 0

Hence, roots are imaginary.

(ii) 2x² – 7x + 3 = 0 (Here, a = 2, b = – 7, c = 3)

∴ D = b² – 4ac = 49 – 24 = 25 > 0 and a perfect square

Hence, roots are real and rational.

(iii) x² – 5x – 2 = 0. (Here, a = 1, b = – 5, c = – 2)

∴ D = b² – 4ac = 25 + 8 = 33 > 0 and not a perfect square

Hence, roots are real and irrational.

(iv) 4x² – 4x + 1 = 0 (Here, a = 4, b = – 4, c = 1)

∴ D = b² – 4ac = 16 – 16 = 0. Hence, roots are real and equal.