NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3

NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3 helps students extend their problem-solving expertise in quadratic equations with two new methods. The exercise goes beyond the traditional factorisation method for solving the quadratic equations with the method of completing the square & the quadratic formula. These methods are mostly used when the quadratic equations don’t have a common term to form two binomials.

Here, you can find NCERT Solutions Class 10 Maths Chapter 4 Exercise 4.3 PDF, along with some important key concepts of the exercise in brief. 

1.0Download NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3: Free PDF

2.0Introduction to Solutions of Quadratic Equations:

Let us first recall what solutions of a quadratic equation are before proceeding to the steps of solving quadratic equations. The solutions of any given quadratic equation are the variable values (mainly x), which set the equation to zero. They are determined when the quadratic equations have been expressed in their standard form. That is:

$a{x}^2+bx+c=0$

In this expression, a and b are coefficients of the variable x, and c is the constant term. Solutions of quadratic equations are also referred to as the roots or the zeroes of the equations.

3.0Key Concepts of Exercise 4.3

Exercise 4.3 will look forward to the new methods of solving quadratic equations, understanding which will equip students with extra tools to find the zeroes of equations. Let’s explore important key concepts of these methods: 

Completing the Square: 

Completing the square method requires a proper approach to solving the quadratic equation, which includes the following: 

• Divide the Equation by a: The first and most important step of this method is dividing the given quadratic equation by the coefficient of x², that is, “a”. This is performed to set the coefficient of x² equal to 1 so that these equations are easier to solve.
• Form a Perfect square: The next step of the method is to make the equation a perfect square trinomial. This is done by adding and subtracting the square of half of the coefficient of x.
• Simplify the Equation: After finishing the square, rewrite the equation to create the product of binomials. Equalise these polynomials to zero to solve for the value of x. This equalisation gives rise to another method of solving quadratic equations, the direct formula or the quadratic formula for the equations. 

The Quadratic Formula: 

The expression derived from the method of completing the square forms a generalised formula for the quadratic equation $a{x}^2+bx+c=0$. Which can be expressed as: 

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

In the formula, b² – 4ac can never be negative if the roots of a quadratic equation are real values. This generalised formula is also known as the quadratic formula for any equation. The formula is a useful tool in solving complex quadratic equations, which require long-form solutions of the equations. 

4.0Exercise 4.3: Overview 

• The first couple of questions of exercise 4.3 includes solving the quadratic equations by completing the square method. 
• The Exercise also involves the solution of numerical problems of solving quadratic equations for roots by the quadratic formula.
• Last but not least, the exercise also has some real-life problems to be solved with the aforementioned methods.

To thrive in solving quadratic equations and build up your knowledge, start practising with the help of NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3.

5.0NCERT Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3: Detailed Solutions 

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) 3x² - 4√3x + 4 = 0

(iii) 2x² - 6x + 3 = 0

Solution:

(i) 2x² - 3x + 5 = 0

Here a = 2, b = -3 and c = 5

Therefore, discriminant D = b² - 4ac

= (-3)² - 4 × 2 × 5

= 9 - 40 = -31

As D < 0

Hence, no real root.

(ii) 3x² - 4√3x + 4 = 0

Here a = 3, b = -4√3 and c = 4

Therefore, discriminant D = b² - 4ac

= (-4√3)² - 4(3)(4) = 48 - 48 = 0

As D = 0

Hence, two equal real roots.

Now, the roots are

= (-b ± √D) / 2a = (4√3 ± 0) / (2 × 3) = 2/√3

Hence, the roots are 2/√3 and 2/√3.

(iii) 2x² - 6x + 3 = 0

a = 2, b = -6 and c = 3

Discriminant D = b² - 4ac

= (-6)² - 4(2)(3)

= 36 - 24 = 12

=> D > 0

Hence, roots are distinct and real.

The roots are

x = (-b ± √D) / 2a

= (6 ± √12) / (2 × 2) = (6 ± 2√3) / 4 = (3 ± √3) / 2

Therefore, the roots are (3 + √3) / 2 and (3 - √3) / 2.

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

Solution:

(i) 2x² + kx + 3 = 0

Here a = 2, b = k and c = 3

Therefore, D = b² - 4ac

= k² - 4 × 2 × 3 = k² - 24

Two roots will be equal if D = 0

=> k² - 24 = 0

=> k = ±√24

=> k = ±2√6

(ii) kx(x - 2) + 6 = 0

or kx² - 2kx + 6 = 0

Here, a = k, b = -2k and c = 6

Therefore, D = b² - 4ac

= (-2k)² - 4(k)(6) = 4k² - 24k

Two roots will be equal if D = 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'. Hence, k = 6.

3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and area is 800 m²? If so, find its length and breadth.

Solution:

Let x be the breadth and 2x be the length of the rectangle.

x × 2x = 800

=> 2x² = 800

=> x² = 400 = (20)²

=> x = 20

Hence, the rectangle is possible and it has breadth = 20 m and length = 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.

Solution:

Let the age of one friend be x years.

The age of the other friend will be (20 - x) years.

4 years ago, age of first friend = (x - 4) years and age of second friend = (20 - x - 4) = (16 - x) years

Given that,

(x - 4)(16 - x) = 48

16x - 64 - x² + 4x = 48

-x² + 20x - 112 = 0

x² - 20x + 112 = 0

Here, a = 1, b = -20 and c = 112

Therefore, D = b² - 4ac

= (-20)² - 4(1)(112)

= 400 - 448 = -48

As D < 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 m²? If so, find its length and breadth.

Solution:

Perimeter of the rectangular park = 80 m

=> Length + Breadth of the park = 80/2 m = 40 m

Let the breadth be x metres, then length = (40 - x) m.

Here, x < 40.

x × (40 - x) = 400 [Area of the park]

=> -x² + 40x - 400 = 0

=> x² - 40x + 400 = 0

=> (x - 20)² = 0

=> x = 20

Thus, we have length = breadth = 20 m.

Therefore, the park is in the shape of a square having 20 m side.

6.0Benefits of studying NCERT Class 10 Maths Chapter 4 Quadratic Equations Exercise 4.3

• Exercise 4.3 focuses on solving quadratic equations using the Quadratic Formula, which is a fundamental concept in algebra.
• NCERT exercises are aligned with the CBSE syllabus, making them crucial for board exam preparation.
• Many competitive exams (JEE, NTSE, Olympiads) include questions based on quadratic equations.
• Exercise 4.3 helps in mastering essential techniques for such exams.