Question

1. Find the nature of the roots of the following quadratic equations. If the real roots exist,
find them:
(ii) 3x2 - 4 V3 x + 4 = 0
(1) 2x2 – 3x +5=0
(iii) 2x2 - 6x +3=0​

Answer 1

Step-by-step explanation:

Given:-

(ii) 3x^2 - 4 V3 x + 4 = 0

(1) 2x^2 – 3x +5=0

(iii) 2x^2 - 6x +3=0

To find:-

Find the nature of the roots of the following quadratic equations. If the real roots exist,

find them

Solution:-

i)Given quadratic equation is 3x^2-4√3 x+4 = 0

On comparing with the standard quadratic equation ax^2+bx+c=0

a = 3

b=-4√3

c=4

To know the nature of the roots we have to find the value of the discriminant.

The discriminant (D)= b^2-4ac

D= (-4√3)^2-4(3)(4)

D=48-48

D=0

Since the value of discriminant is equal to zero then the given equation has equal and real roots

Now roots of the given equation

by Quadratic formula

x=[-b±√(b^2-4ac)]/2a

x=-b/2a (since D = 0)

=>x = -(-4√3)/2×3

=>x=4√3/6

=>x = 2√3/3

The roots are 2√3/3 and 2√3/3

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i)Given quadratic equation is 2x^2 – 3x +5=0

On comparing with the standard quadratic equation ax^2+bx+c=0

a = 2

b=-3

c=5

To know the nature of the roots we have to find the value of the discriminant.

The discriminant (D)= b^2-4ac

D=> (-3)^2-4(2)(5)

D=>9-40

D=-31

D<0

Since the value of discriminant is less than zero then the given equation has no real roots i.e. imaginary .

No real roots exist .

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iii)Given quadratic equation is 2x^2 - 6x +3=0

On comparing with the standard quadratic equation ax^2+bx+c=0

a = 2

b= -6

c=3

To know the nature of the roots we have to find the value of the discriminant.

The discriminant (D)= b^2-4ac

D=> (-6)^2-4(2)(3)

D=> 36-24

D=>12

D> 0

Since the value of discriminant is equal to zero then the given equation has distinct and real roots.

Now roots of the given equation

by Quadratic formula

x=[-b±√(b^2-4ac)]/2a

=>x = [-(-6)±√12]/(2×2)

=>x= [6±√12)/4

=>x= (6±2√3)/4

=>x= 2(3±√3)/4

=>x= (3±√3)/2

=>x = (3+√3)/2 and (3-√3)/2

The roots are (3+√3)/2 and (3-√3)/2

Used formulae:-

• The standard quadratic equation ax^2+bx+c=0

• Quadratic formula

x=[-b±√(b^2-4ac)]/2a

• The discriminant of ax^2+bx+c = 0 is
• D = b^2-4ac
• If D > 0 the equation has real and distinct roots.
• If D =0 the equation has real and equal roots.
• If D > 0 the equation has no real roots.i.e.imaginary.
• The real roots exist if D≥0