Chapter 4: Quadratic Equations

Overview

This page provides an Overview of Chapter Wise Notes for CBSE Class 10. Key topics covered: Exam Weightage & Blueprint, â° Last 24-Hour Checklist, Concepts & Solving Methods ★★★★★, Nature of Roots (Discriminant) 🔥🔥🔥, Solved Examples (Board Marking Scheme), Previous Year Questions (PYQs), Exam Strategy & Mistake Bank, Concept Mastery Quiz 🎯. All content is aligned with the NCERT syllabus for Board Exam preparation.

Board Exam Focused Notes, Formulas, and PYQs

Exam Weightage & Blueprint

Total: 4-6 Marks

This chapter is part of the Algebra Unit (20 Marks Total). Focus on nature of roots and word problems.

Question Type	Marks	Frequency	Focus Topic
MCQ	1	High	Nature of Roots (Discriminant)
Short Answer	2 or 3	Medium	Solving by Factorisation/Formula
Long Answer	4 or 5	Medium	Word Problems (Speed/Age/Area)

â° Last 24-Hour Checklist

Standard Form: $ax^2 + bx + c = 0, a \neq 0$.
Discriminant: $D = b^2 - 4ac$.
Quadratic Formula: $x = \frac{-b \pm \sqrt{D}}{2a}$.

Nature of Roots: $D > 0, D = 0, D < 0$.
Speed Formula: Time = Distance / Speed.
Dimension Check: Length cannot be negative.

Concepts & Solving Methods ★★★★★

Standard Form: A quadratic equation in variable $x$ is of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$.

1. Method of Factorisation

Split the middle term $bx$ such that the product of the two parts equals $ac$.

Example: $2x^2 - 5x + 3 = 0$.
Split $-5x$ into $-2x$ and $-3x$ because $(-2)(-3) = 6 = (2)(3)$.

2. Quadratic Formula

The roots of $ax^2 + bx + c = 0$ are given by:

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

Nature of Roots (Discriminant) 🔥🔥🔥

The Discriminant is $D = b^2 - 4ac$. It determines the nature of roots without solving.

Value of D ($b^2 - 4ac$)	Nature of Roots	Roots
D > 0	Two Distinct Real Roots	$\frac{-b \pm \sqrt{D}}{2a}$
D = 0	Two Equal Real Roots	$-\frac{b}{2a}, -\frac{b}{2a}$
D < 0	No Real Roots	Imaginary

⚠️ Common Mistake: If the question asks "Find $k$ for real roots", use $D \ge 0$ (combining $D>0$ and $D=0$). Don't just use $D>0$.

Quadratic Root Finder

Enter coefficients for $ax^2 + bx + c = 0$

Solved Examples (Board Marking Scheme)

Q1. Find the discriminant of $2x^2 - 4x + 3 = 0$ and find the nature of roots. (2 Marks)

Step 1: Identify Coefficients 0.5 Mark

Step 2: Calculate Discriminant 1 Mark

Step 3: Conclusion 0.5 Mark

Q2. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides. (4 Marks)

Step 1: Form Equation 1 Mark

Step 2: Simplify 1 Mark

Step 3: Solve 1.5 Marks

Step 4: Conclusion 0.5 Mark

Previous Year Questions (PYQs)

2023 (2 Marks): Find the value of $k$ for which the equation $2x^2 + kx + 3 = 0$ has two equal roots.
Ans: For equal roots, $D = 0 \Rightarrow b^2 - 4ac = 0$.
$k^2 - 4(2)(3) = 0 \Rightarrow k^2 = 24 \Rightarrow k = \pm 2\sqrt{6}$.

2020 (3 Marks): Solve for x: $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$.
Ans: Split $7x$ into $2x + 5x$. Roots are $-\sqrt{2}, -\frac{5}{\sqrt{2}}$.

2019 (4 Marks): A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed.
Ans: Eq: $\frac{360}{x} - \frac{360}{x+5} = 1$. Solving gives $x = 40$ km/h ($x=-45$ rejected).

Exam Strategy & Mistake Bank

Mistake Bank 🚨

Formula Error: Forgetting the $\pm$ in the quadratic formula or calculating $b^2 - 4ac$ incorrectly (sign errors).

Rejecting Roots: Forgetting to explain WHY a negative root is rejected (e.g., "Speed cannot be negative").

Square Root: writing $\sqrt{16} = \pm 4$. Note: $\sqrt{16}$ is 4. The equation $x^2=16$ gives $x=\pm 4$.

Scoring Tips 🏆

Identify Form: Always bring equation to $ax^2+bx+c=0$ first.

Units: In word problems, don't forget units (km/h, m, years) in the final answer.

Check D: Calculate $D$ first. If $D$ is a perfect square, calculation is likely correct.

Concept Mastery Quiz 🎯

Test your readiness for the board exam.

1. The quadratic equation $ax^2 + bx + c = 0$ has no real roots if:

2. The roots of the equation $x^2 - 3x - 10 = 0$ are:

3. Which of the following is NOT a quadratic equation?