NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations Exercise 4.4

NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations Exercise 4.4 are provided to assist students in their Class 10 exam preparations. These solutions are designed by subject experts to explain how to solve quadratic equations that arise in different real-life scenarios. The solutions use methods like factorization, quadratic formula, and completing the square, helping students understand and apply these methods effectively.

Exercise 4.4 focuses on solving word problems involving quadratic equations, and it helps students understand how to form equations from given situations and solve them algebraically. By practicing this exercise, students will learn how to approach different types of problems and apply quadratic equations to solve them.

NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations Exercise 4.4

Quadratic Equations

Medium

Q.

$\begin{array}{l}\text{Find the nature of the roots of the following quadratic}\\ \text{equations. If the real roots exist, find them.}\\ \text{(i) }2{\mathrm{x}}^2-3\mathrm{x}+5=0\text{ (ii)\hspace{0.17em}\hspace{0.17em}}3{\mathrm{x}}^2-4\sqrt{3}\mathrm{x}+4=0\\ \text{(iii) }2{\mathrm{x}}^2-6\mathrm{x}+3=0\end{array}$

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Quadratic Equations

Medium

Q.

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

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Quadratic Equations

Medium

Q.

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.

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Quadratic Equations

Medium

Q.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

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Quadratic Equations

Medium

Q.

Find the value of k for which the roots of the equation $$\rm 3 x^2-10 x+k=0$$​ are reciprocal of each other.

[CBSE - 2019]

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These solutions are aligned with the latest CBSE syllabus and provide step-by-step explanations, making them an excellent resource for students to prepare confidently for board exams.

Q1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

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

• Solution: Here $a = 2, b = -3, c = 5$.

• Discriminant ($D$): $D = b^2 - 4ac = (-3)^2 - 4(2)(5) = 9 - 40 = -31$.

• Explanation: Since $D < 0$, the roots are not real.

• Answer: No real roots exist.

(ii) $3x^2 - 4\sqrt{3}x + 4 = 0$

• Solution: Here $a = 3, b = -4\sqrt{3}, c = 4$.

• Discriminant ($D$): $D = (-4\sqrt{3})^2 - 4(3)(4) = (16 \times 3) - 48 = 48 - 48 = 0$.

• Explanation: Since $D = 0$, the equation has two equal real roots.

• Finding Roots: $x = \frac{-b}{2a} = \frac{4\sqrt{3}}{2(3)} = \frac{2\sqrt{3}}{3} = \frac{2}{\sqrt{3}}$.

• Answer: Two equal real roots: $\frac{2}{\sqrt{3}}, \frac{2}{\sqrt{3}}$.

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

• Solution: Here $a = 2, b = -6, c = 3$.

• Discriminant ($D$): $D = (-6)^2 - 4(2)(3) = 36 - 24 = 12$.

• Explanation: Since $D > 0$, the equation has two distinct real roots.

• Finding Roots: $x = \frac{-(-6) \pm \sqrt{12}}{2(2)} = \frac{6 \pm 2\sqrt{3}}{4} = \frac{3 \pm \sqrt{3}}{2}$.

• Answer: Two distinct real roots: $\frac{3+\sqrt{3}}{2}$ and $\frac{3-\sqrt{3}}{2}$.

Q2. Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots:

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

• Logic: For equal roots, $D = 0$.

• Solution: $k^2 - 4(2)(3) = 0 \implies k^2 - 24 = 0$.

• Calculation: $k^2 = 24 \implies k = \pm \sqrt{24} = \pm 2\sqrt{6}$.

• Answer: $k = \pm 2\sqrt{6}$.

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

• Simplify: $kx^2 - 2kx + 6 = 0$. Here $a = k, b = -2k, c = 6$.

• Logic: $D = (-2k)^2 - 4(k)(6) = 0$.

• Calculation: $4k^2 - 24k = 0 \implies 4k(k - 6) = 0$.

• Explanation: $k$ cannot be $0$ (it would not be quadratic), so $k - 6 = 0$.

• Answer: $k = 6$.

Q3. Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is $800\text{ m}^2$? If so, find its length and breadth.

• Logic: Let Breadth $= x$. Length $= 2x$.

• Equation: $x \times 2x = 800 \implies 2x^2 = 800 \implies x^2 = 400$.

• Nature Check: $x^2 - 400 = 0$. Here $D = 0^2 - 4(1)(-400) = 1600 > 0$. Yes, it is possible.

• Solution: $x = \sqrt{400} = 20$.

• Answer: Breadth = 20m, Length = 40m.

Q4. 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.

• Logic: Friend 1 $= x$. Friend 2 $= 20 - x$.

• 4 Years Ago: $(x - 4)(20 - x - 4) = 48 \implies (x - 4)(16 - x) = 48$.

• Equation: $16x - x^2 - 64 + 4x = 48 \implies x^2 - 20x + 112 = 0$.

• Nature Check: $D = (-20)^2 - 4(1)(112) = 400 - 448 = -48$.

• Explanation: Since $D < 0$, there are no real solutions.

• Answer: This situation is not possible.

Q5. Is it possible to design a rectangular park of perimeter 80m and area $400\text{ m}^2$? If so, find its length and breadth.

• Logic: $2(L + B) = 80 \implies L + B = 40$. Let $L = x, B = 40 - x$.

• Equation: $x(40 - x) = 400 \implies 40x - x^2 = 400 \implies x^2 - 40x + 400 = 0$.

• Nature Check: $D = (-40)^2 - 4(1)(400) = 1600 - 1600 = 0$. Yes, it is possible.

• Solution: $(x - 20)^2 = 0 \implies x = 20$.

• Answer: Length = 20m, Breadth = 20m (It is a square park).

FAQs: Class 10 Maths Chapter 4 – Quadratic Equations Exercise 4.4

Q1. What is the focus of Exercise 4.4?
Answer:
Exercise 4.4 focuses on solving word problems using quadratic equations. It helps students understand how to form quadratic equations based on the given real-life situations and solve them algebraically.

Q2. How do I solve word problems in Exercise 4.4?
Answer:
To solve word problems:

1. Read the problem carefully and identify the given data.

2. Translate the problem into a quadratic equation.

3. Use factorization or the quadratic formula

   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$x=2a−b±b2−4ac​​ to find the roots.

4. Interpret the solutions in the context of the problem.

Q3. What methods are used to solve quadratic equations in this exercise?
Answer:
The methods used to solve quadratic equations in this exercise are:

• Factorization Method: Decomposing the quadratic equation into factors.

• Quadratic Formula: Using the formula

  $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$x=2a−b±b2−4ac​​ to find the roots.

• Completing the Square: A method used to transform the quadratic equation into a perfect square trinomial.

Q4. Can quadratic equations be applied to real-life problems?
Answer:
Yes, quadratic equations are commonly used to solve problems in geometry, motion, business, and finance, such as calculating areas, projectile motion, profit and loss, and determining unknown values from given conditions.

Q5. How do NCERT Solutions help with exam preparation?
Answer:
These solutions provide clear, detailed explanations of how to form and solve quadratic equations in various scenarios. By practicing these problems, students will improve their problem-solving skills and gain confidence in solving word problems in Class 10 exams.