Solution:
Q i. x2 –3x + 4 = 0
Solution:-
(i)
The equation x2 – 3x + 4 = 0 hasno real roots.
D = b2 – 4ac
= (-3)2 – 4(1)(4)
= 9 – 16 < 0
Hence, the roots are imaginary.
Q ii. 2x2 +x – 1 = 0
Solution:-
(ii)
The equation 2x2 + x – 1 = 0has two real and distinct roots.
D = b2 – 4ac
= 12 – 4(2) (-1)
= 1 + 8 > 0
Hence, the roots are real and distinct.
Q iii. 2x2 –6x + 9/2 = 0
Solution:-
(iii)
The equation 2x2 – 6x +(9/2) = 0 has real and equal roots.
D = b2 – 4ac
= (-6)2 – 4(2) (9/2)
= 36 – 36 = 0
Hence, the roots are real and equal.
Q iv. 3x2 – 4x + 1 = 0
Solution:-
(iv)
The equation 3x2 – 4x + 1 = 0has two real and distinct roots.
D = b2 – 4ac
= (-4)2 – 4(3)(1)
= 16 – 12 > 0
Hence, the roots are real and distinct.
Q v. (x + 4)2 – 8x = 0
Solution:-
(v)
The equation (x + 4)2 – 8x = 0has no real roots.
Simplifying the above equation,
x2 + 8x + 16 – 8x = 0
x2 + 16 = 0
D = b2 – 4ac
= (0) – 4(1) (16)< 0
Hence, the roots are imaginary.
Q vi. (x – √2)2 –2(x + 1) = 0
Solution:-
(vi)
The equation (x – √2)2 – √2(x+1)=0 hastwo distinct and real roots.
Simplifying the above equation,
x2 – 2√2x + 2 – √2x – √2 = 0
x2 – √2(2+1)x + (2 – √2) = 0
x2 – 3√2x + (2 – √2) = 0
D = b2 – 4ac
= (– 3√2)2 – 4(1)(2 – √2)
= 18 – 8 + 4√2 > 0
Hence, the roots are real and distinct.
Q vii. √2 x2 –(3/√2)x+ 1/√2 = 0
Solution:-
(vii)
The equation √2x2 – 3x/√2 + ½= 0 has two real and distinct roots.
D = b2 – 4ac
= (- 3/√2)2 – 4(√2) (½)
= (9/2) – 2√2 > 0
Hence, the roots are real and distinct.
Q viii. x (1 – x) – 2 = 0
Solution:-
(viii)
The equation x (1 – x) – 2 = 0 has no realroots.
Simplifying the above equation,
x2 – x + 2 = 0
D = b2 – 4ac
= (-1)2 –4(1)(2)
= 1 – 8 < 0
Hence, the roots are imaginary.
Q ix. (x – 1) (x + 2) + 2 = 0
Solution:-
(ix)
The equation (x – 1) (x + 2) + 2 = 0 has tworeal and distinct roots.
Simplifying the above equation,
x2 – x + 2x – 2 + 2 = 0
x2 + x = 0
D = b2 – 4ac
= 12 –4(1)(0)
= 1 – 0 > 0
Hence, the roots are real and distinct.
Q x. (x + 1) (x – 2) + x = 0
Solution:-
(x)
The equation (x + 1) (x – 2) + x = 0 has tworeal and distinct roots.
Simplifying the above equation,
x2 + x – 2x – 2 + x = 0
x2 – 2 = 0
D = b2 – 4ac
= (0)2 – 4(1) (-2)
= 0 + 8 > 0
Hence, the roots are real and distinct.