Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3 NCERT Solutions

NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.3 focuses on solving quadratic equations using the method of completing the square. This exercise introduces a more systematic approach where equations are rewritten in a perfect square form to find their roots.

It is an important concept in the CBSE Class 10 Maths syllabus and helps in understanding how algebraic expressions can be transformed for easier solving.

The step-by-step solutions explain how to rearrange terms, create perfect squares, and simplify expressions correctly. Practising these questions improves clarity in algebraic manipulation and prepares a strong foundation for solving equations using the quadratic formula.

NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.3

1. Find the nature of the roots of the following quadratic equation. If the real roots exist, find them:

(i) 2x2 – 3x + 5 = 0

(ii) 3x 2 -4√3x+4 = 0

(iii) 2x2– 6x + 3 = 0

Answer:

We know that the quadratic equation ax 2 +bx+c = 0 has
(a) Two distinct real roots, if b 2 -4ac > 0,
(b) Two equal real roots, if b 2 -4ac = 0,
(c) No real roots, if b 2 -4ac < 0.

(i) 2x 2 -3x+5 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax 2 +bx+c = 0,

we get a = 2, b = -3 and c = 5
Then, b 2 -4ac = (-3) 2 -4×2×5 = 9-40 = -31 < 0
Hence, the given quadratic equation has no real roots.

(ii) 3x 2 -4√3x+4 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax 2 +bx+c = 0

we get a = 3, b = -4√3 and c = 4
Then, b 2 -4ac = (-4√3) 2 -4×3×4 = 48-48 = 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) 2x 2 -6x+3 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax2+bx+c = 0,
we get a = 2, b = -6 and c = 3
then, b 2 -4ac = (-6) 2 -4×2×3 = 36-24 = 12 > 0
Hence, the given quadratic equation has two distinct real roots.

Applying the quadratic formulato find roots,

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

(i) 2x 2 +kx+3 = 0

Answer:
Comparing the given quadratic equation with the general form of quadratic equation ax 2 +bx+c = 0, we get
a = 2, b = k and c = 3
Now the given quadratic equation have two equal roots if
b 2 -4ac = 0

(k)2-4×2×3 = 0
k 2 -24 = 0
k 2 = 24
k = ±√24
k = ± 2√6

Therefore, the required value of k is ± 2√6

(ii)  kx(x-2)+6 = 0

Answer:
The given quadratic equation can be written as
kx 2 -2kx+6 = 0….(1)
Comparing the quadratic equation (1) with the general form of quadratic equation ax 2 +bx+c = 0 we get
a = k, b = -2k and c = 6

Now the given quadratic equation have two equal roots if
b 2 -4ac = 0
(-2k) 2 -4×k×6 = 0
4k 2 -24k = 0
4k(k-6) = 0
k(k-6) = 0
k = 0
Or, k-6 = 0
k = 6

If k = 0, then the equation will not have x 2 and x, which is not possible because the given equation is quadratic.
Therefore, the required value of k is 6.

3. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is 800 m 2? If so, find its length and breadth.

Answer:
Let the breadth of the mango grove be x m and the length is 2x m.
Area = length × breadth
= x × 2x
= 2x2 m 2
Then by the given condition,
2x 2 = 800
x 2 = 400
x = ±√400
x = ±20
Since length cannot be negative, then x ≠ -20
Hence, it is possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m 2, and its breadth is 20 m, and length is 20×2 = 40 m

4. Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Answer:
Let the age of 1st friend is x.
Then the age of 2nd friend is (20-x)
Four years ago their age was (x-4) and (20-x-4)
Then, using the given condition, we have
(x-4)(16-x) = 48
16x – 64 – x 2 +4x = 48
20x – x 2 – 64 – 48 = 0
x 2 – 20x + 112 = 0……(1)

Now, comparing the above quadratic equation with the general form of quadratic equation ax 2 +bx+c = 0 we get
a = 1, b = -20 and c = 112
then, b 2 -4ac = (-20) 2 -4×1×112 = 400-448 = -48 > 0

Therefore, no real root is possible for this equation, and hence, this situation is not possible.

5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m 2? If so, find its length and breadth.

Answer:
Let the length and breadth of the park be “l” and “b”
Area of rectangle = 2(l + b)
2(l + b) = 80
l + b = 40
b = 40 – l
Then, Area = l(40 – l)
Then by the given condition,
l(40 – l) = 400
40l  – l 2 -400 = 0
l 2 – 40l + 400 = 0….(1)

Now, comparing the above quadratic equation with the general form of quadratic equation ax 2 +bx+c = 0 we get
a = 1, b = -40 and c = 400
Then, b 2 -4ac = (-40) 2 -4×1×400 = 1600-1600 = 0
As the quadratic equation has two equal roots, the given situation is possible.

Therefore, length of park, l = 20 m

And breadth of park, b = 40 - l = 40 - 20 = 20 m.

How to Score Better in Class 10 Maths Exam?

Scoring well in Class 10 Maths requires clear concepts, regular practice, and a focus on accuracy and answer presentation. To score better, you should:

• Build Strong Concepts:

Focus on understanding concepts in Class 10 Maths instead of memorising steps, as this helps in solving application-based questions.

• Work on Weak Areas:

Focus on difficult topics from the Class 10 Maths syllabus instead of skipping them to avoid losing marks.

• Revise Formulas Daily:

Regular revision of PW Class 10 Maths MIQs helps avoid calculation mistakes in exams.

• Practise Regularly:

Solve all CBSE Class 10 NCERT questions multiple times to strengthen your basics and improve accuracy.

• Solve Previous Year Papers:

Practising CBSE Class 10 Maths previous year questions (PYQs) helps you understand question patterns and important topics.