Find the nature of roots of the following quadratic equations:(i) 5x (

1.	Find the nature of roots of the following quadratic equations:(i) 5x (x - 2) + 6 = 0(ii) 12x2 - 4\(\sqrt{15}\)x + 5 = 0(iii) x2 - x + 2 = 0
Answer» (i) The given equation is 5x (x - 2) + 6 = 0 ⇒ 5x² - 10x + 6 = 0 This is of the form ax² + bx + c = 0,where a = 5,b = -10 and c = 6 ∴ Discriminant, D = b² - 4ac = (-10)² - 4 x 5 x 6 = 100 - 120 = - 20 <0 Hence, the given equation has real and equal roots. (ii) The given equation is 12x² - 4\(\sqrt{15}\)x + 5 = 0 This is of the form ax² + bx + c = 0,where a = 12,b = -4\(\sqrt{15}\) and c = 5 ∴ Discriminant, D = b² - 4ac = (- 4\(\sqrt{15}\))² - 4 x 12 x 5 = 240 - 240 = 0 Hence, the given equation has real and equal roots. (iii) The given equation is x² - x + 2 = 0 This is of the form ax² + bx + c = 0,where a = 1,b = -1 and c = 2 ∴ Discriminant, D = b² - 4ac = (- 1)² - 4 x 1 x 2 = 1 - 8 = -7 < 0 Hence, the given equation has no real roots.

Find the nature of roots of the following quadratic equations:(i) 5x (x - 2) + 6 = 0(ii) 12x2 - 4\(\sqrt{15}\)x + 5 = 0(iii) x2 - x + 2 = 0

Answer»

(i) The given equation is

5x (x - 2) + 6 = 0

⇒ 5x² - 10x + 6 = 0

This is of the form ax² + bx + c = 0,where a = 5,b = -10 and c = 6

∴ Discriminant, D = b² - 4ac = (-10)² - 4 x 5 x 6 = 100 - 120 = - 20 <0

Hence, the given equation has real and equal roots.

(ii) The given equation is 12x² - 4\(\sqrt{15}\)x + 5 = 0

This is of the form ax² + bx + c = 0,where a = 12,b = -4\(\sqrt{15}\) and c = 5

∴ Discriminant, D = b² - 4ac = (- 4\(\sqrt{15}\))² - 4 x 12 x 5 = 240 - 240 = 0

Hence, the given equation has real and equal roots.

(iii) The given equation is x² - x + 2 = 0

This is of the form ax² + bx + c = 0,where a = 1,b = -1 and c = 2

∴ Discriminant, D = b² - 4ac = (- 1)² - 4 x 1 x 2 = 1 - 8 = -7 < 0

Hence, the given equation has no real roots.