Determine the nature of the roots for the following quadratic equation

1.	Determine the nature of the roots for the following quadratic equations (i) 15.x2 + 11.x + 2 = 0 (ii) x2 – x – 1 = 0 (iii) √2t2 – 3t + 3√2 = 0 (iv) 9y2 – 6√2y + 2 = 0 (v) 9a2 b2 x2 – 24abcdx + 16c2 d2 = 0, a ≠ 0, b ≠ 0
Answer» (i) 15x² + 11x + 2 = 0 comparing with ax² + bx + c = 0. Here a = 15, 6 = 11, c = 2. Δ = b² – 4ac = 11² - 4 x 15 x 2 = 121 – 120 = 1 > 1. ∴ The roots are real and unequal. (ii) x² – x – 1 = 0, Here a = 1, b = -1, c = -1. Δ = b² – 4ac = (-1)² – 4 x 1 x -1 = 1 + 4 = 5 > 0. ∴ The roots are real and unequal. (iii) √2t² – 3t + 3√2 = 0 Here a = √2, b = -3, c = 3√2 Δ = b² – 4ac = (-3)²– 4 x √2 x 3√2 = 9 – 24 = -15 < 0. ∴ The roots are not real. (iv) 9y² – 6√2y + 2 = 0 a = 9, b = 6√2, c = 2 Δ = b² – 4ac = (6√2)² – 4 x 9 x 2 = 36 x 2 – 72 = 72 – 72 = 0 ∴ The roots are real and equal. (v) 9a² b² x² – 24abcdx + 16c² d² = 0 Δ = b² – 4ac = (-24abcd)² – 4 x 9a² b² x 16c² d² = 576a² b² c² d² – 576a² b² c² d² = 0 ∴ The roots are real and equal.

Determine the nature of the roots for the following quadratic equations (i) 15.x2 + 11.x + 2 = 0 (ii) x2 – x – 1 = 0 (iii) √2t2 – 3t + 3√2 = 0 (iv) 9y2 – 6√2y + 2 = 0 (v) 9a2 b2 x2 – 24abcdx + 16c2 d2 = 0, a ≠ 0, b ≠ 0

Answer»

(i) 15x² + 11x + 2 = 0 comparing with ax² + bx + c = 0.

Here a = 15, 6 = 11, c = 2.

Δ = b² – 4ac

= 11² - 4 x 15 x 2

= 121 – 120

= 1 > 1.

∴ The roots are real and unequal.

(ii) x² – x – 1 = 0,

Here a = 1, b = -1, c = -1.

Δ = b² – 4ac

= (-1)² – 4 x 1 x -1

= 1 + 4 = 5 > 0.

∴ The roots are real and unequal.

(iii) √2t² – 3t + 3√2 = 0

Here a = √2, b = -3, c = 3√2

Δ = b² – 4ac

= (-3)²– 4 x √2 x 3√2

= 9 – 24 = -15 < 0.

∴ The roots are not real.

(iv) 9y² – 6√2y + 2 = 0

a = 9, b = 6√2, c = 2

Δ = b² – 4ac =

(6√2)² – 4 x 9 x 2

= 36 x 2 – 72

= 72 – 72 = 0

∴ The roots are real and equal.

(v) 9a² b² x² – 24abcdx + 16c² d² = 0

Δ = b² – 4ac

= (-24abcd)² – 4 x 9a² b² x 16c² d²

= 576a² b² c² d² – 576a² b² c² d² = 0

∴ The roots are real and equal.