Class 10 Math Chapter 4 Quadratic Equations

March 18, 2026

Class 10 Math Chapter 4 Quadratic Equations – Complete NCERT Solutions

Introduction

Quadratic equations are everywhere – from calculating areas to solving age problems. A quadratic equation is of the form $a x^{2} + b x + c = 0$ , where $a \neq 0$ . In this chapter, we learn how to solve quadratic equations using factorisation, the quadratic formula, and by understanding the nature of roots.

Key Formulas

Standard form: $a x^{2} + b x + c = 0$
Quadratic formula:

x=−b±b2−4ac2a

Discriminant (D): $D = b^{2} - 4 a c$
- If $D > 0$ : two distinct real roots
- If $D = 0$ : two equal real roots
- If $D < 0$ : no real roots

Solved Examples from NCERT

Example 1: John and Jivanti’s marbles problem. Equation: $x^{2} - 45 x + 324 = 0$ . Roots give John’s marbles = 9, Jivanti’s = 36.

Example 2: Toys production cost problem. Equation: $x^{2} - 55 x + 750 = 0$ . Roots give toys produced = 25.

Example 3: Solve $2 x^{2} - 5 x + 3 = 0$ by factorisation. Solution: $(2 x - 3) (x - 1) = 0$ . Roots = $\frac{3}{2}, 1$ .

Example 4: Solve $6 x^{2} - x - 2 = 0$ . Solution: $(3 x - 2) (2 x + 1) = 0$ . Roots = $\frac{2}{3}, - \frac{1}{2}$ .

Example 6: Prayer hall problem. Equation: $2 x^{2} + x - 300 = 0$ . Roots = 12, -12.5. Breadth = 12 m, Length = 25 m.

Example 7: Discriminant of $2 x^{2} - 4 x + 3 = 0$ . $D = - 8 < 0$ . No real roots.

Example 8: Circular park pole problem. Equation: $x^{2} + 7 x - 60 = 0$ . Roots = 5, -12. Valid solution = 5 m from gate B.

Exercise Solutions (Step by Step)

Each NCERT exercise question is solved with clear steps. For example:

Exercise 4.2 (i): Solve $x^{2} - 3 x - 10 = 0$ . Factorisation: $(x - 5) (x + 2) = 0$ . Roots = 5, -2.

15 FAQs with Solutions

Q: What is a quadratic equation? A: An equation of the form $a x^{2} + b x + c = 0$ .
Q: Solve $x^{2} - 2 x - 8 = 0$ . A: Roots = 4, -2.
Q: What is the discriminant of $x^{2} + 4 x + 4 = 0$ ? A: $D = 0$ . Roots are equal.
Q: Solve $2 x^{2} + x - 6 = 0$ . A: Roots = $\frac{3}{2}, - 2$ .
Q: Solve $100 x^{2} - 20 x + 1 = 0$ . A: Roots = $\frac{1}{10}$ .
Q: Solve $x^{2} + 7 x + 10 = 0$ . A: Roots = -2, -5.
Q: Solve $x^{2} - 3 = 0$ . A: Roots = $\sqrt{3}, - \sqrt{3}$ .
Q: Solve $2 x^{2} - 3 x + 5 = 0$ . A: No real roots.
Q: Solve $2 x^{2} - 6 x + 3 = 0$ . A: Roots = $\frac{3 \pm \sqrt{3}}{2}$ .
Q: Solve $3 x^{2} - 2 x - \frac{1}{3} = 0$ . A: Roots = $\frac{1}{3}$ .
Q: Solve train speed problem (Exercise 4.1). A: Speed = 48 km/h.
Q: Solve Rohan’s age problem. A: Rohan = 12 years.
Q: Solve integers product = 306. A: Integers = 17, 18.
Q: Solve rectangular plot area = 528 m². A: Breadth = 12 m, Length = 25 m.
Q: What is the quadratic formula used for? A: To find roots of quadratic equations when factorisation is not easy.

Conclusion

Quadratic equations are a cornerstone of algebra. By mastering factorisation, the quadratic formula, and discriminant analysis, you’ll be able to solve real‑life problems with confidence. Practice each NCERT exercise thoroughly with these step‑by‑step solutions.

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