Find the nature of roots of the following quadratic equations:(i) 2x2

1.	Find the nature of roots of the following quadratic equations:(i) 2x2 - 8x + 5 = 0(ii) 3x2 - 2√6x + 2 = 0(iii) 5x2 - 4x + 1 = 0
Answer» (i) The given equation is 2x² - 8x + 5 = 0 This is of the form ax² + bx + c = 0, where a = -8 and c = 5 ∴ Discriminant, D = b² - 4ac = (-8)² - 4 x 2 x 5 = 64 - 40 = 24 > 0 Hence, the given equation has real and unequal roots. (ii) The given equation is 3x² - 2√6x + 2 = 0 This is of the form ax² + bx + c, where a = 3, b = -2√6 and c = 2. ∴ Discriminant, D = b² - 4ac = (-2√6)² - 4 x 3 x 2 = 24 - 24 = 0 Hence, the given equation has real and equal roots. (iii) The given equation is 5x²- 4x + 1 = 0 This is of the form ax² + bx + c = 0, where a = 5,b = -4 and c = 1 ∴ Discriminant, D = b² - 4ac = (-4)² - 4 x 5 x 1 = 16 - 20 = -4 < 0 Hence, the given equation has no real roots.

Find the nature of roots of the following quadratic equations:(i) 2x2 - 8x + 5 = 0(ii) 3x2 - 2√6x + 2 = 0(iii) 5x2 - 4x + 1 = 0

Answer»

(i) The given equation is 2x² - 8x + 5 = 0

This is of the form ax² + bx + c = 0, where a = -8 and c = 5

∴ Discriminant, D = b² - 4ac = (-8)² - 4 x 2 x 5 = 64 - 40 = 24 > 0

Hence, the given equation has real and unequal roots.

(ii) The given equation is 3x² - 2√6x + 2 = 0

This is of the form ax² + bx + c, where a = 3, b = -2√6 and c = 2.

∴ Discriminant, D = b² - 4ac = (-2√6)² - 4 x 3 x 2 = 24 - 24 = 0

Hence, the given equation has real and equal roots.

(iii) The given equation is 5x²- 4x + 1 = 0

This is of the form ax² + bx + c = 0, where a = 5,b = -4 and c = 1

∴ Discriminant, D = b² - 4ac = (-4)² - 4 x 5 x 1 = 16 - 20 = -4 < 0

Hence, the given equation has no real roots.