Question

write the discriminant and nature of roots of the follwing :

1. x²-3x+4=0
2. x²+3x+12=0
3. 2x²-x+1=0
4. x²+4=0
5. x²-5x=0​

Answer 1

comparing all eq. from standard form of quadratic equation ax²+bx+c=0

1) a= 1 , b= -3 ,c= 4

D=b²-4ac= 9-4×4 = -7

D<0 it have imaginary roots

2) a= 1, b=3 ,c= 12

D=b²-4ac= 9- 4×12 = -39

D<0 it have imaginary roots

3) a= 2 , b= -1 ,c= 1

D=b²-4ac= 1- 4×2×1 = -7

D<0 it have imaginary roots

4) a= 1 , b= 0 ,c= 4

D=b²-4ac= 0 - 4×4 = -16

D<0 it have imaginary roots

5)a= 1 , b= 0 ,c= -5

D=b²-4ac= 0 - 4×5 = -20

D<0 it have imaginary roots

Answer 2

Step-by-step explanation:

Commonly , we can find the nature of solution by using the formula:

del = b² - 4ac

On comparison with ax² + bx + c = 0, we get:

1. a=1, b=(-3), c = 4
2. a=1, b=3, c=12
3. a=2,b=(-1),c=1
4. a=1,b=0, c=4
5. a=1, b= (-5), c=0

Solution:

1. del = (-3)² - 4(1)(4)

=9 - 16

del = -7 <0

Nature - not real

2. del = (3)²-4(1)(12)

= 9 - 48

del = (-39)<0

Nature - not real

3. del = (-1)² - 4(2)(1)

= 1 - 8

del = (-7) < 0

Nature - not real

4. del = 0² - 16

del = -16 < 0

Nature - not real

5. del = (-5)² - 4(1)(0)

= 25 - 0

del = 25 > 0

Nature - Real and unequal