Quadratic Equations Class 10
Download the PDF containing Quadratic Equations Class 10 notes and a mind map for chapter revision. Use the formulas, explanations and worked examples below to understand factorisation, the quadratic formula, the discriminant and applications.
Finding the dimensions of a rectangle or working with consecutive integers can lead to an equation containing the square of an unknown. Class 10 Quadratic Equations teaches you to recognise these equations, solve them and interpret their roots in context. The first step is always to simplify the equation and identify its coefficients correctly.
These Quadratic Equations Class 10 notes provide a structured revision guide. Read the explanations for understanding, use the mind map to recall the connections between formulas, and practise questions without looking at the solutions. This approach helps you choose a method rather than apply a formula mechanically.
What Is a Quadratic Equation?
A quadratic equation in one variable can be written in the standard form ax2 + bx + c = 0, where a ≠ 0. Here, a, b and c are real coefficients. The highest power of the variable is 2 after simplification.
For example, 2x2 − 5x + 3 = 0 is quadratic, with a = 2, b = −5 and c = 3. The equation x2 = 9 is also quadratic: its standard form is x2 − 9 = 0. A missing linear term means b = 0.
Simplify Before Identifying the Degree
An equation may display squared terms that cancel. For example, (x + 1)2 = x2 + 5 simplifies to 2x − 4 = 0, so it is linear. Decide whether an equation is quadratic only after bringing all terms to one side and simplifying.
Quadratic Equations Class 10 Formula Sheet
| Concept | Formula | Condition or Use |
|---|---|---|
| Standard form | ax2 + bx + c = 0 | a ≠ 0 |
| Discriminant | D = b2 − 4ac | Determines the nature of the roots. |
| Quadratic formula | x = [−b ± √(b2 − 4ac)] / (2a) | Gives real roots when D ≥ 0. |
| Repeated root | x = −b/(2a) | When D = 0. |
| Sum of roots | α + β = −b/a | Useful for checking roots. |
| Product of roots | αβ = c/a | Useful for checking roots. |
Discriminant and Nature of Roots
A root is a value of the variable that satisfies the equation. Calculate the discriminant when you need to determine the nature of the roots without solving the equation fully.
| Discriminant | Nature of Roots | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x2 − 5x + 6 = 0 |
| D = 0 | Two equal real roots | x2 − 6x + 9 = 0 |
| D < 0 | No real roots | x2 + 1 = 0 |
Equal roots represent one distinct solution repeated twice. For x2 − 6x + 9 = 0, the factorisation is (x − 3)2 = 0, so both roots are 3.
Solving Quadratic Equations by Factorisation
Factorisation is convenient when the quadratic expression splits into simple linear factors. To split the middle term of ax2 + bx + c, find two numbers whose sum is b and whose product is ac.
Example 1: Solve 2x² − 7x + 3 = 0
Product ac = 2 × 3 = 6
Required numbers are −6 and −1.
2x2 − 6x − x + 3 = 0
2x(x − 3) − (x − 3) = 0
(2x − 1)(x − 3) = 0
2x − 1 = 0 or x − 3 = 0
x = 1/2 or 3
The zero-product property applies because the product equals zero: at least one factor must be zero.
Solving by the Quadratic Formula
Use the quadratic formula Class 10 when factors are not easy to identify. Write the equation in standard form, record a, b and c with their signs, and calculate the discriminant before substituting into the formula.
Example 2: Solve x² − 4x + 1 = 0
a = 1, b = −4, c = 1
D = (−4)2 − 4 × 1 × 1
D = 12
x = [4 ± √12]/2
x = [4 ± 2√3]/2
x = 2 + √3 or 2 − √3
Leave the answer in exact surd form unless a decimal approximation is requested.
Finding an Unknown Value Using Equal Roots
Example 3: Find k for Equal Roots
Question: Find k if x2 − 8x + k = 0 has equal real roots.
Equal roots require D = 0.
(−8)2 − 4 × 1 × k = 0
64 − 4k = 0
4k = 64
k = 16
The equation becomes (x − 4)2 = 0, confirming the repeated root 4.
Quadratic Equations Class 10 Word Problems
Begin a word problem by defining the variable and expressing the other quantities in terms of it. Form an equation, solve it and check each root against the conditions. A valid algebraic root may be unsuitable as a length, speed or number of objects.
Example 4: Dimensions of a Rectangle
Question: A rectangle has area 48 cm2. Its length is 2 cm greater than its breadth. Find its dimensions.
Let the breadth be x cm.
Length = (x + 2) cm
x(x + 2) = 48
x2 + 2x − 48 = 0
(x + 8)(x − 6) = 0
x = −8 or x = 6
Reject −8 because a breadth must be positive.
Breadth = 6 cm
Length = 8 cm
Check: 6 × 8 = 48 cm2, and the length exceeds the breadth by 2 cm.
Using the Quadratic Equations Class 10 Mind Map
Use the Quadratic Equations Class 10 mind map to connect standard form, coefficients, solving methods and root conditions. Recall the quadratic formula, then explain how the sign of the discriminant affects its square-root term.
For active revision, cover the formula section and write it from memory. Next, solve one factorisation question and one discriminant question without referring to the notes. Use the detailed explanations again whenever you cannot justify a step.
Extra Questions and MCQs with Answers
-
Solve x2 − 9x + 20 = 0.
Answer: x = 4 or 5. -
Find the discriminant of 3x2 − 2x + 1 = 0.
Answer: D = −8, so there are no real roots. -
Solve x2 − 10x + 25 = 0.
Answer: Both roots are 5. -
Find k if x2 + kx + 9 = 0 has equal roots.
Answer: k = 6 or −6.
Quick MCQ Check
1. Which condition gives two distinct real roots?
A. D < 0 B. D = 0 C. D > 0 D. a = 0
Answer: C.
2. For x² + 5x = 0, which statement is correct?
A. The only root is −5
B. The roots are 0 and −5
C. The roots are 0 and 5
D. There are no real roots
Answer: B. Factorise as x(x + 5) = 0.
NCERT Practice and Chapter Revision
Use these Quadratic Equations Class 10 notes PDF resources alongside your NCERT exercises. Match the question wording with your textbook when searching for Exercise 4.1, 4.2 or 4.3, because exercise organisation can differ between editions.
Practise identifying quadratic equations, finding roots and determining their nature before attempting application problems. For important questions and verified previous year questions, show the selected method and check the result. A clear equation and correct interpretation are especially important in word problems.
Related Worksheets and Notes
Apply these concepts using the Quadratic Equations Class 10 Worksheet with Solutions . Attempt each question independently before checking the working.
Revise related algebra in the CBSE Class 10 Maths Notes and Mind Maps collection .
Frequently Asked Questions
Does this Quadratic Equations Class 10 PDF contain notes and a mind map?
Yes. The PDF combines notes for understanding the chapter with a mind map for quick revision. Read the notes first, then use the map to recall formulas and connections before practising questions.
Why must a be non-zero in a quadratic equation?
If a = 0, the squared term disappears. The equation then has degree at most 1 and is no longer quadratic. The quadratic formula also requires a non-zero denominator 2a.
Can b or c be zero?
Yes. Only a must be non-zero. For example, x2 − 16 = 0 has b = 0, while x2 − 5x = 0 has c = 0. Both are quadratic equations.
How do I choose between factorisation and the quadratic formula?
Use factorisation when the factors are easy to identify. Use the quadratic formula when splitting the middle term is difficult. If only the nature of roots is asked, calculate the discriminant without finding both roots.
What condition gives equal roots?
Equal real roots occur when b2 − 4ac = 0. The repeated root is −b/(2a). For example, x2 − 4x + 4 = 0 has the repeated root 2.
Does a negative discriminant mean the equation has no roots?
It means the equation has no real roots. Write “no real roots” when working within the real number system rather than claiming there are no roots of any kind.
Can a quadratic equation have irrational roots?
Yes. For example, x2 − 2 = 0 has roots √2 and −√2. With rational coefficients, a positive discriminant that is not a square of a rational number gives irrational real roots.
Why should I not divide x² + 5x = 0 by x immediately?
Dividing by x assumes x ≠ 0 and loses the valid root 0. Instead, factorise as x(x + 5) = 0 to obtain both roots: 0 and −5.
Why is a negative root sometimes rejected in a word problem?
A negative value may be incompatible with the quantity represented, such as a length or speed. Reject a root because it violates the problem’s conditions, not simply because it is negative. Some problems involving integers allow negative answers.
Can completing the square also solve quadratic equations?
Yes. For example, x2 − 6x + 5 = 0 becomes (x − 3)2 = 4, giving x = 1 or 5. It also explains the origin of the quadratic formula. Follow the method specified in your question when one is given.
How can I check my roots?
Substitute each root into the original equation. You can also check that their sum is −b/a and product is c/a. In an application problem, verify the original conditions as well.
What mistakes should I avoid in the quadratic formula?
Record negative coefficients in brackets, use −b rather than b, and divide the entire numerator by 2a. Include both the plus and minus cases. Simplify surds carefully and round only when requested.
Where can I practise Quadratic Equations Class 10 questions with solutions?
Visit the Quadratic Equations Class 10 worksheet with solutions . Use it after revising the notes and mind map to practise formula selection, calculations and applications.
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