CBSE Class 10 Mathematics • Chapter 4
Quadratic Equations Class 10 Worksheet with Solutions
Practise factorisation, the quadratic formula and the nature of roots. Build your understanding with clear explanations, worked examples, MCQs and word problems, then check your progress using the extra questions and answers below.
A Quadratic Equations Class 10 worksheet helps you move from recognising an equation to solving it accurately. The chapter involves more than remembering a formula: you must simplify expressions, identify coefficients with their signs and decide whether the roots are meaningful in a given situation.
Use the worksheet available on this page alongside these revision notes and original practice questions. This worksheet on Quadratic Equations Class 10 can support classroom practice, homework and self-revision. Attempt each problem independently before checking the solution, and substitute your roots into the original equation whenever possible.
What Is a Quadratic Equation?
A quadratic equation in one variable x can be written in the standard form ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0. After simplification, its highest power of the variable must be 2.
Quadratic Equation Examples
- x2 − 5x + 6 = 0
- 2x2 + 3x − 2 = 0
- x2 − 9 = 0
- 3x2 + 6x = 0
- 4x2 = 0
The coefficients b or c may be zero. However, a cannot be zero, because the x2 term is necessary for a quadratic equation.
Simplify Before Deciding the Degree
An equation containing x2 is not always quadratic. For example:
(x + 1)2 = x2 + 5
x2 + 2x + 1 = x2 + 5
2x − 4 = 0
The squared terms cancel, leaving a linear equation.
Quadratic Equations Class 10 Formulas
For ax2 + bx + c = 0, begin by identifying a, b and c. The following formula sheet brings together the main relationships used in Class 10 quadratic equations questions.
| Concept | Formula |
|---|---|
| Standard form | ax2 + bx + c = 0, a ≠ 0 |
| Discriminant | D = b2 − 4ac |
| Quadratic formula for real roots, when D ≥ 0 | x = [−b ± √(b2 − 4ac)] / (2a) |
| Repeated root, when D = 0 | x = −b/(2a) |
| Sum of roots α and β | α + β = −b/a |
| Product of roots α and β | αβ = c/a |
| A quadratic equation with roots α and β | x2 − (α + β)x + αβ = 0 |
Discriminant and Nature of Roots
The discriminant tells you the nature of the roots without requiring you to calculate them individually.
| Discriminant | Nature of Roots | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x2 − 5x + 6 = 0; D = 1 |
| D = 0 | Two equal real roots, or one repeated real root | x2 − 6x + 9 = 0; D = 0 |
| D < 0 | No real roots | x2 + x + 1 = 0; D = −3 |
Methods for Solving Class 10 Quadratic Equations
Factorisation
Write the quadratic expression as a product of two linear factors. Set each factor equal to zero. This method is convenient when the factors are easy to identify.
Quadratic Formula
Substitute a, b and c into the formula. Check the discriminant first to determine whether real roots exist. This method is useful when factorisation is less straightforward.
Completing the Square
Rearrange the equation into a squared expression. For example, x2 + 6x + 5 = 0 becomes (x + 3)2 = 4. Follow the methods required by your textbook.
Checking the Roots
Substitute each root into the original equation. In word problems, also check restrictions such as positive dimensions, valid ages or non-zero denominators.
Class 10 Quadratic Equations Important Questions with Solutions
Example 1: Solve by Factorisation
Question: Solve 2x2 − 7x + 3 = 0.
Split the middle term:
2x2 − 6x − x + 3 = 0
2x(x − 3) − 1(x − 3) = 0
(2x − 1)(x − 3) = 0
Therefore:
2x − 1 = 0 or x − 3 = 0
x = 1/2 or x = 3
Check x = 3:
2(3)2 − 7(3) + 3 = 0
Check x = 1/2:
2(1/2)2 − 7(1/2) + 3 = 0
Example 2: Apply the Quadratic Formula
Question: Solve x2 − 4x − 1 = 0.
Identify the coefficients:
a = 1
b = −4
c = −1
Calculate the discriminant:
D = (−4)2 − 4(1)(−1)
D = 16 + 4
D = 20
Apply the formula:
x = [4 ± √20] / 2
x = [4 ± 2√5] / 2
x = 2 + √5 or x = 2 − √5
Since D > 0, the equation has two distinct real roots.
Example 3: Find the Value of k for Equal Roots
Question: Find k if x2 − 8x + k = 0 has equal roots.
Equal roots require D = 0.
(−8)2 − 4(1)(k) = 0
64 − 4k = 0
4k = 64
k = 16
The equation becomes:
x2 − 8x + 16 = 0
(x − 4)2 = 0
The repeated root is x = 4.
Quadratic Equations Class 10 Word Problems
Case Study: Dimensions of a Rectangular Garden
A rectangular garden has an area of 84 square metres. Its length is 5 metres more than its width. Find the dimensions.
Let the width be x metres, where x > 0.
The length is (x + 5) metres.
Use area = length × width:
x(x + 5) = 84
x2 + 5x − 84 = 0
x2 + 12x − 7x − 84 = 0
x(x + 12) − 7(x + 12) = 0
(x − 7)(x + 12) = 0
Therefore:
x = 7 or x = −12
Reject x = −12 because a garden cannot have a negative width. The width is 7 metres and the length is 12 metres.
Check the area: 7 × 12 = 84 square metres.
Check the difference: 12 − 7 = 5 metres.
Similar quadratic equation questions can involve consecutive integers, products of numbers, ages or speed and time. Define the unknown quantity before forming the equation, and explain why any unsuitable root is rejected.
Quadratic Equations Class 10 MCQs with Answers
MCQ 1: Identify a Quadratic Equation
Which equation is quadratic?
A. 3x + 2 = 0
B. x2 − 4x + 3 = 0
C. x3 + 1 = 0
D. 2x − 7 = 0
Show Answer
Answer: B. Its degree is 2 and the coefficient of x2 is non-zero.
MCQ 2: Nature of Roots
What is the nature of the roots of x2 + 2x + 5 = 0?
A. Two distinct real roots
B. Equal real roots
C. No real roots
D. One root is zero
Show Answer
Answer: C.
D = 22 − 4(1)(5) = −16.
A negative discriminant means there are no real roots.
MCQ 3: Find the Roots
What are the roots of x2 − 9 = 0?
A. 3 and 3
B. −3 and −3
C. 3 and −3
D. 0 and 9
Show Answer
Answer: C. (x − 3)(x + 3) = 0 gives x = 3 or x = −3.
MCQ 4: Condition for Equal Roots
For ax2 + bx + c = 0 to have equal roots, which condition must hold?
A. b2 − 4ac > 0
B. b2 − 4ac = 0
C. b2 − 4ac < 0
D. a = 0
Show Answer
Answer: B. Equal real roots occur when the discriminant is zero. The condition a ≠ 0 must also hold for the equation to remain quadratic.
Quadratic Equations Class 10 Extra Questions
Attempt these original practice questions as a short revision test. Show your working before checking the answers.
- Solve x2 − 11x + 24 = 0 by factorisation.
- Solve 3x2 − 12 = 0.
- Determine the nature of the roots of 2x2 + 3x + 5 = 0.
- Find k if x2 − 10x + k = 0 has equal roots.
- Solve x2 + 2x − 2 = 0 using the quadratic formula.
- The product of two consecutive positive integers is 72. Form a quadratic equation and find the integers.
Check the Practice Answers
- x = 3 or x = 8.
- x = 2 or x = −2.
- D = −31; there are no real roots.
- k = 25; the repeated root is x = 5.
- x = −1 + √3 or x = −1 − √3.
- x(x + 1) = 72 gives x2 + x − 72 = 0. The positive integers are 8 and 9.
How to Use This Worksheet with NCERT Exercises and PYQs
When practising Class 10 Quadratic Equations Exercise 4.1, begin with recognising quadratic equations and forming them from situations. For Exercise 4.2 and Exercise 4.3, follow the questions and method instructions in your own NCERT edition, since exercise organisation can differ between editions.
Use these examples to strengthen your approach to Quadratic Equations Class 10 NCERT Exemplar problems, sample paper questions and PYQs. When using a Quadratic Equations Class 10 previous year questions PDF with answers, check the board and examination year. The extra questions written here are original practice questions.
Common Mistakes to Avoid
- Applying the formula before writing the equation in standard form.
- Forgetting the negative sign in b when identifying coefficients.
- Dividing only the square-root term by 2a instead of the entire numerator.
- Losing the zero root by dividing an equation by x.
- Writing “no roots” instead of “no real roots” when D < 0.
- Accepting a negative length, age or speed without checking the context.
Related Quadratic Equations Notes, Solutions and Worksheets
FAQs on Quadratic Equations Class 10 Worksheet
What is a quadratic equation? Give an example.
A quadratic equation can be written as ax2 + bx + c = 0, where a ≠ 0. For example, x2 − 5x + 6 = 0 is quadratic. Its roots are 2 and 3 because both values satisfy the equation.
What is the quadratic formula for Class 10?
The formula is x = [−b ± √(b2 − 4ac)] / (2a). Identify the coefficients from standard form, including their signs. For real roots, the discriminant b2 − 4ac must be non-negative.
How do I choose between factorisation and the quadratic formula?
Use factorisation when the expression splits into factors easily. For example, x2 − 5x + 6 becomes (x − 2)(x − 3). Use the quadratic formula when the factors are difficult to identify, or when the question requires that method.
What condition must the discriminant satisfy for equal roots?
Equal real roots require b2 − 4ac = 0. For example, x2 − 6x + 9 = 0 has discriminant zero and the repeated root x = 3.
How do I find k when a quadratic equation has equal roots?
Set the discriminant equal to zero and solve for k. For example, x2 − 8x + k = 0 gives 64 − 4k = 0, so k = 16. If k appears in the coefficient of x2, also check that the selected value does not make that coefficient zero.
Does every quadratic equation have real roots?
No. A negative discriminant means there are no real roots. For example, x2 + 1 = 0 has D = −4. Class 10 questions usually ask you to identify this as having no real roots.
Can zero be a root of a quadratic equation?
Yes. For example, x2 − 3x = 0 factors as x(x − 3) = 0. Its roots are 0 and 3. Dividing by x would lose the zero root, so use factorisation instead.
Why do we reject some roots in quadratic word problems?
A root may satisfy the equation but fail the conditions of the situation. For example, a negative value cannot represent a garden's width. State the restriction and explain why the unsuitable root is rejected.
What is the difference between polynomial zeroes and equation roots?
A zero of p(x) is a value that makes p(x) = 0. A root of the equation p(x) = 0 is the same value. Thus, the zeroes of x2 − 5x + 6 are the roots of x2 − 5x + 6 = 0.
Which Quadratic Equations Class 10 important questions should I practise?
Practise recognising quadratic equations, finding roots by factorisation and the formula, determining the nature of roots, finding unknown coefficients and forming equations from word problems. Include questions that require you to justify your answer.
How should I use a Quadratic Equations Class 10 PDF with solutions?
Solve the questions before checking the solutions. Compare your equation formation, signs, calculations and root selection. Reattempt incorrect questions without looking at the answers to check whether you have understood the method.
Are CBSE Chapter 4 and Maharashtra Board Maths 1 Chapter 2 the same resource?
Both cover quadratic equations, but their textbook questions and exercise numbering differ. For Maharashtra Board practice sets, use the Class 10 Maths 1 Chapter 2 Quadratic Equations Solutions . For CBSE, follow Chapter 4 in your NCERT textbook.
Can I use this worksheet alongside RS Aggarwal or ML Aggarwal?
Yes, you can use it for additional concept practice. Match questions to your board, textbook and assigned syllabus. This page does not provide exercise-by-exercise solutions for RS Aggarwal or ML Aggarwal.
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