Class 10 NCERT Maths Solution Exercise-4.4,Quadratic Equations.

Class X Ncert Maths

Exercise 4.4

Question 1:

Find the nature of the roots of the following quadratic equations.

If the real roots exist, find them;

(I) 2x² −3x + 5 = 0

(II)

(III) 2x² − 6x + 3 = 0

We know that for a quadratic equation ax² + bx + c = 0, discriminant is b² − 4ac.

(A) If b² − 4ac > 0 → two distinct real roots

(B) If b² − 4ac = 0 → two equal real roots

(C) If b² − 4ac < 0 → no real roots

(I) 2x² −3x + 5 = 0

Comparing this equation with ax² + bx + c = 0, we obtain

a = 2, b = −3, c = 5

Discriminant = b² − 4ac = (− 3)² − 4 (2) (5) = 9 − 40

= −31

As b² − 4ac < 0,

Therefore, no real root is possible for the given equation.

(II)

Comparing this equation with ax² + bx + c = 0, we obtain

Discriminant

= 48 − 48 = 0

As b² − 4ac = 0,

Therefore, real roots exist for the given equation and they are equal to each other.

And the roots will be

 

Therefore, the roots are  

.

(III) 2x² − 6x + 3 = 0

Comparing this equation with ax² + bx + c = 0, we obtain

a = 2, b = −6, c = 3

Discriminant = b² − 4ac = (− 6)² − 4 (2) (3)

= 36 − 24 = 12

As b² − 4ac > 0,

Therefore, distinct real roots exist for this equation as follows.

Therefore, the roots are 

.

Question 2:

Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(I) 2x² + kx + 3 = 0

(II) kx (x − 2) + 6 = 0

We know that if an equation ax² + bx + c = 0 has two equal roots, its discriminant
(b² − 4ac) will be 0.

(I) 2x² + kx + 3 = 0

Comparing equation with ax² + bx + c = 0, we obtain

a = 2, b = k, c = 3

Discriminant = b² − 4ac = (k)²− 4(2) (3)

= k² − 24

For equal roots,

Discriminant = 0

k² − 24 = 0

k² = 24

 

(II) kx (x − 2) + 6 = 0

or kx² − 2kx + 6 = 0

Comparing this equation with ax² + bx + c = 0, we obtain

a = k, b = −2k, c = 6

Discriminant = b² − 4ac = (− 2k)² − 4 (k) (6)

= 4k² − 24k

For equal roots,

b² − 4ac = 0

4k² − 24k = 0

4k (k − 6) = 0

Either 4k = 0 or k = 6 = 0

k = 0 or k = 6

However, if k = 0, then the equation will not have the terms ‘x²’ and ‘x’.

Therefore, if this equation has two equal roots, k should be 6 only.

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