Determine the nature of roots of the following quadratic equations. i

1.	Determine the nature of roots of the following quadratic equations. i. x2 – 4x + 4 = 0 ii. 2y2 – 7y + 2 = 0 iii. m2 + 2m + 9 = 0
Answer» i. x² – 4x + 4= 0 Comparing the above equation with ax² + bx + c = 0, we get a = 1, b = -4, c = 4 ∴ ∆ = b² – 4ac = (- 4)² – 4 × 1 × 4 = 16 – 16 ∴ ∆ = 0 ∴ Roots of the given quadratic equation are real and equal. ii. 2y² – 7y + 2 = 0 Comparing the above equation with ay² + by + c = 0, we get a = 2, b = -7, c = 2 ∴ ∆ = b² – 4ac = (- 7)² – 4 × 2 × 2 = 49 – 16 ∴ ∆ = 33 ∴ ∆ > 0 ∴ Roots of the given quadratic equation are real and unequal. iii. m² + 2m + 9 = 0 Comparing the above equation with am² + bm + c = 0, we get a = 1, b = 2, c = 9 ∴ ∆ = b² – 4ac = (2)² – 4 × 1 × 9 = 4 – 36 ∴ ∆ = -32 ∴ ∆ < 0 ∴ Roots of the given quadratic equation are not real.

Determine the nature of roots of the following quadratic equations. i. x2 – 4x + 4 = 0 ii. 2y2 – 7y + 2 = 0 iii. m2 + 2m + 9 = 0

Answer»

i. x² – 4x + 4= 0

Comparing the above equation with

ax² + bx + c = 0, we get

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

∴ ∆ = b² – 4ac

= (- 4)² – 4 × 1 × 4

= 16 – 16

∴ ∆ = 0

∴ Roots of the given quadratic equation are real and equal.

ii. 2y² – 7y + 2 = 0

Comparing the above equation with

ay² + by + c = 0, we get

a = 2, b = -7, c = 2

∴ ∆ = b² – 4ac

= (- 7)² – 4 × 2 × 2

= 49 – 16

∴ ∆ = 33

∴ ∆ > 0

∴ Roots of the given quadratic equation are real and unequal.

iii. m² + 2m + 9 = 0

Comparing the above equation with

am² + bm + c = 0, we get

a = 1, b = 2, c = 9

∴ ∆ = b² – 4ac

= (2)² – 4 × 1 × 9

= 4 – 36

∴ ∆ = -32

∴ ∆ < 0

∴ Roots of the given quadratic equation are not real.