Class 10 · Chapter 4 · Algebra unit (20 of 80 marks)

Quadratic Equations Class 10: notes and important questions

Q: What is in CBSE Class 10 Maths Chapter 4, Quadratic Equations?

The topics are: Standard form of a quadratic equation, Solution by factorisation, Nature of roots, Solution by quadratic formula, Situational problems. This page has revision notes for each, and 49 board-style practice questions with step-marked solutions.

Q: How many marks is Quadratic Equations worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Q: Where do students lose marks in Quadratic Equations?

Not rejecting the negative or impossible root in word problems (negative speed, age, length). Fix: write 'x = −60 is rejected as speed cannot be negative'; the scheme gives a mark for it. Using the formula without writing a, b, c and D first. Fix: list a = , b = , c = , D = b² − 4ac = … before substituting; these are step marks.

Q: Are the Quadratic Equations important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Q: Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Revision notes, 10 free important questions with step-marked answers, and a four-step route from the basics to 95+ through 49 board-style questions.

Practise this chapter Join freeFree account, no card: +5 practice questions on every chapter, with instant feedback.

49 questions
15 multiple choice, 4 assertion–reason, 11 very short answer, 10 short answer, 6 long answer, 3 case study
About 12 hours to master

Algebra unit: 20 of 80 theory marks (Polynomials, Pair of Linear Equations, Quadratic Equations, Arithmetic Progressions).

NCERT textbook: Chapter 4 on ncert.nic.in (PDF)

Revision notes

Quadratic Equations — revision notes

1. Standard form

$a x^{2} + b x + c = 0$ with $a \neq 0$ . Always expand and simplify first: $(x - 1)^{2} = x^{2} + 2$ looks quadratic but is linear.

2. Solving by factorisation (splitting the middle term)

Find two numbers whose product is $a c$ and whose sum is $b$ . E.g. $6 x^{2} + 7 x - 20$ : $a c = - 120 = 15 \times (- 8)$ , so $6 x^{2} + 15 x - 8 x - 20 = (2 x + 5) (3 x - 4)$ ; roots $- \frac{5}{2}, \frac{4}{3}$ .

3. Quadratic formula

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}, D = b^{2} - 4 a c$

4. Nature of roots

$D > 0$ : two distinct real roots.
$D = 0$ : two equal real roots, each $- \frac{b}{2 a}$ .
$D < 0$ : no real roots.

5. Situational problems

Speed–time: time $= \frac{distance}{speed}$ ; form “time difference” equations.
Work/taps: rates add: $\frac{1}{x} + \frac{1}{y} = \frac{1}{T}$ .
Always reject roots that are negative or otherwise impossible, and say why.

Worked example 1

Solve $x^{2} + 2 x - 4 = 0$ : $D = 4 + 16 = 20$ , $x = \frac{- 2 \pm 2 \sqrt{5}}{2} = - 1 \pm \sqrt{5}$ .

Worked example 2

Find $k$ so that $x^{2} - k x + 16 = 0$ has equal roots: $k^{2} - 64 = 0 \Rightarrow k = \pm 8$ .

Worked example 3

A car covers 300 km; if it went 10 km/h faster it would take 1 hour less. $\frac{300}{x} - \frac{300}{x + 10} = 1 \Rightarrow x^{2} + 10 x - 3000 = 0 \Rightarrow (x + 60) (x - 50) = 0$ , so $x = 50$ km/h.

Common errors

Dividing both sides by $x$ and losing the root $x = 0$ .
Using $b$ instead of $- b$ in the formula, or dividing only the root term by $2 a$ .
Forgetting to rearrange to “ $= 0$ ” before factorising.
Not rejecting inadmissible roots in word problems.

Board-exam tips

Write $a, b, c$ and $D$ explicitly before using the formula — step marks are awarded for these.
For “find $k$ ” questions, state the condition ( $D = 0$ , $D \geq 0$ , …) before solving.
Check the answer in the original word problem (e.g. times and speeds make sense).

Topics in this chapter: Standard form of a quadratic equation · Solution by factorisation · Nature of roots · Solution by quadratic formula · Situational problems.

Formulas for this chapter: Chapter 4 on the Class 10 formula sheet (free PDF and A3 poster).

Quadratic Equations Class 10: important questions

10 original questions in the board paper's question types, from 1-mark MCQs to a case study, each with a step-marked worked answer. Try each one first. 5 worked answers open here; the other 5 open with a free account (no card).

Download these questions (PDF) 10 questions, 5 worked answers · 5 pages · 166 KB

Section A style · Multiple choice and assertion–reason (1 mark each)

Q1·1 mark·Multiple choiceStandard form of a quadratic equation

For which value of $k$ is $x = 2$ a root of $k x^{2} - 3 x - 10 = 0$ ?

Show worked answer

$\begin{aligned} k (4) - 6 - 10 = 0 \\ \Rightarrow & 4 k = 16 \\ \Rightarrow & k = 4 \end{aligned}$ A1
Correct option: (b) $4$

Q2·1 mark·Multiple choiceNature of roots

Which statement about the roots of $9 x^{2} + 12 x + 4 = 0$ is true?

Q3·1 mark·Assertion–reasonSolution by factorisation

In the following question, a statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option.

Assertion (A): The roots of $2 x^{2} - 7 x + 3 = 0$ are $3$ and $\frac{1}{2}$ .

Reason (R): The roots of $a x^{2} + b x + c = 0$ always add up to $\frac{b}{a}$ .

Section B style · Short answer (2 marks each)

Q4·2 marks·Short answerSolution by quadratic formula

Find the roots of $x^{2} - 3 \sqrt{5} x + 10 = 0$ .

Show worked answer

Answer:

2 \sqrt{5}

and

\sqrt{5}

$D = (3 \sqrt{5})^{2} - 4 (1) (10) = 45 - 40 = 5$ ; $x = \frac{3 \sqrt{5} \pm \sqrt{5}}{2}$ M1
$x = 2 \sqrt{5}$ or $x = \sqrt{5}$ A1

Q5·2 marks·Short answerNature of roots

Find the value of $k$ for which $(k - 1) x^{2} + 4 x + 1 = 0$ has two equal roots.

Section C style · Short answer (3 marks each)

Q6·3 marks·Short answerSolution by factorisation

Solve for $x$ : $\frac{2}{x} - \frac{3}{x + 2} = \frac{1}{4}$ , where $x \neq 0, - 2$ .

Show worked answer

Answer:

x = 2

x = - 8

$\begin{aligned} \frac{2 (x + 2) - 3 x}{x (x + 2)} = \frac{1}{4} \\ \Rightarrow & 4 (4 - x) = x^{2} + 2 x \end{aligned}$ M1
$\begin{aligned} x^{2} + 6 x - 16 = 0 \\ \Rightarrow & (x + 8) (x - 2) = 0 \end{aligned}$ A1
$x = 2$ or $x = - 8$ A1

Q7·3 marks·Short answerSituational problems

A cyclist rides $48$ km at a steady speed. Had the speed been $4$ km/h more, the ride would have taken $1$ hour less. Find the cyclist's speed.

Section D style · Long answer (5 marks each)

Q8·5 marks·Long answerSituational problems

The perimeter of a rectangular field is $82$ m and its area is $400 m^{2}$ . Form a quadratic equation and solve it to find the length and breadth of the field.

Show worked answer

Answer:

25

m by

16

Length $+$ breadth $= 41$ ; with breadth $x$ m, length $= (41 - x)$ m M1
$\begin{aligned} x (41 - x) = 400 \\ \Rightarrow & x^{2} - 41 x + 400 = 0 \end{aligned}$ A1
$(x - 16) (x - 25) = 0$ M1
$x = 16$ or $x = 25$ A1
Either way the sides are $25$ m and $16$ m: length $25$ m, breadth $16$ m A1

Q9·5 marks·Long answerSituational problems

Two pipes together fill a tank in $6$ hours. Alone, the larger pipe fills it $5$ hours faster than the smaller one. How long does each pipe take alone?

Section E style · Case study (4 marks)

Q10·4 marks·Case studySituational problems

Photo frame. A photograph measuring $20$ cm by $12$ cm is mounted on a card, leaving a border of the same width $x$ cm all round. The total area of the card is $384 {cm}^{2}$ .

Write the area of the card in terms of $x$ . [1 mark]
Form a quadratic equation in $x$ in standard form. [1 mark]
Find the width of the border and the outer dimensions of the card. [2 marks]

Show worked answer

$(20 + 2 x) (12 + 2 x) {cm}^{2}$ A1
$\begin{aligned} 4 x^{2} + 64 x + 240 = 384 \\ \Rightarrow & x^{2} + 16 x - 36 = 0 \end{aligned}$ A1
$(x + 18) (x - 2) = 0$ ; width cannot be negative, so $x = 2$ cm M1
Card: $24$ cm by $16$ cm (check: $24 \times 16 = 384$ ) A1

Next: practise by Route to 95 step · the revision notes · free MCQs and case studies · all Class 10 assertion–reason questions · the Class 10 syllabus checklist.

Route to 95: four steps

Work through the steps in order. Take each checkpoint closed book, about 1.5 minutes per mark; pass at 80% to move on. Two misses in a row means going back one step.

Step 1

Secure the basics

You can recognise a quadratic, solve it by factorisation or the formula and find the discriminant.

Read first: 1. Standard form; 2. Solving by factorisation (splitting the middle term); 3. Quadratic formula 11 practice questions · checkpoint: 4 questions, 8 marks, pass 80%

Practise step 1

Step 2

Board standard

You can decide the nature of the roots, find k for equal or real roots and set up quadratic word problems from speed, area and number situations.

Read first: 3. Quadratic formula; 4. Nature of roots; 5. Situational problems; Worked examples 1-2 16 practice questions · checkpoint: 4 questions, 10 marks, pass 80%

Practise step 2

Step 3

Full marks on long answers

You can write full-marks 5-mark word problems: equation formed, solved, the impossible root rejected with a reason, answer in context.

Read first: 5. Situational problems; Worked example 3; Board-exam tips 7 practice questions · checkpoint: 3 questions, 13 marks, pass 80%

Practise step 3

Step 4

95+ stretch (HOTS)

You can handle surd coefficients, rational equations and 'for every value of m' discriminant proofs.

Read first: 4. Nature of roots; Common errors 5 practice questions · checkpoint: 3 questions, 10 marks, pass 80%

Practise step 4

Practice questions

Original questions in the board's styles. Multiple-choice answers are checked as you go; for written answers, compare your working with the step mark scheme and record your marks. 10 of the 49 are competency-based (case studies and questions set in a real-life situation): the "Competency-based" button shows just those.

Q1·1 mark·Multiple choiceStandard form of a quadratic equation

Which of the following is a quadratic equation?

(a) $(x + 2)^{2} = x^{2} + 5$
(b) $x (x + 1) + 8 = (x + 2) (x - 2)$
(c) $(x - 3) (2 x + 1) = x (x + 5)$
(d) $x + 3 = 2 x - 7$

Q2·1 mark·Multiple choiceStandard form of a quadratic equation

If $x = - 2$ is a root of $3 x^{2} + 7 x + p = 0$ , then $p$ equals

(a) $26$
(b) $- 2$
(c) $2$
(d) $- 26$

Q3·1 mark·Multiple choiceSolution by factorisation

The roots of $x^{2} - 7 x + 10 = 0$ are

(a) $1, 10$
(b) $- 2, - 5$
(c) $2, 5$
(d) $- 2, 5$

Worked solution·2 marks·Very short answerNature of roots

Find the values of $k$ for which the equation $x^{2} + 2 k x + (k + 6) = 0$ has two equal roots.

Full worked solution Free, no sign-in

Equal roots: $D = (2 k)^{2} - 4 (k + 6) = 0 \Rightarrow k^{2} - k - 6 = 0$ M1
$(k - 3) (k + 2) = 0 \Rightarrow k = 3$ or $k = - 2$ A1

Every question here has a full worked solution. See them all →

Stretch yourself: original algebra problems at Intermediate level (ages 13 to 16), with hints and full solutions: Problem I115 · Problem I122 · Problem I25 · Problem I107. All 29 →

Themed lesson: Fibonacci Day maths lesson · National Mathematics Day maths lesson · Diwali maths lesson (starter, activity and worked answers). All themed maths

Knowledge organiser: Quadratic Equations on one page (key facts, formulas and mistakes to avoid; printable PDF)

Formulas in this topic: Quadratic formula. All formulas A–Z

Where marks are lost in Quadratic Equations

Not rejecting the negative or impossible root in word problems (negative speed, age, length). Fix: write 'x = −60 is rejected as speed cannot be negative'; the scheme gives a mark for it.
Using the formula without writing a, b, c and D first. Fix: list a = , b = , c = , D = b² − 4ac = … before substituting; these are step marks.
Dividing both sides by x (e.g. in x² = 3x) and losing the root x = 0. Fix: bring everything to one side and factorise.
'Find k' questions without stating the condition. Fix: write 'for equal roots, D = 0' (or D ≥ 0 for real roots) before solving for k, and keep both signs of k.
Speed-time equations set up with the wrong sign (time taken is less when speed is more). Fix: write 'time at x − time at (x + 10) = 1' in words first.
Leaving rational equations uncleared or forgetting the restrictions (x ≠ ±1). Fix: state the restriction, multiply through by the LCM and check the roots against it.

Exam strategy: how answers are step-marked · competency-based questions

Quadratic Equations in our sample papers

Once step 3 is passed, test the chapter inside a full timed paper on the 2026-27 pattern (original papers by us, not official CBSE papers).

Sample paper 1 (Board standard): questions 4, 32
Sample paper 2 (Board standard): questions 4, 27, 37
Sample paper 3 (Above board): questions 5, 22
Sample paper 4 (Above board): questions 5, 22, 32, 37
Sample paper 5 (95+ challenge): questions 4, 5, 19, 27

Want it against the clock? Take a timed 30-mark chapter test on Quadratic Equations, new questions each time, or climb the Difficulty ladder, levels 1 to 10, three questions a level (CBSE Pro or the free trial).

Quadratic Equations: common questions

What is in CBSE Class 10 Maths Chapter 4, Quadratic Equations?

The topics are: Standard form of a quadratic equation, Solution by factorisation, Nature of roots, Solution by quadratic formula, Situational problems. This page has revision notes for each, and 49 board-style practice questions with step-marked solutions.

How many marks is Quadratic Equations worth?

It is in the Algebra unit, which carries 20 of the 80 theory marks. CBSE gives marks by unit, not by chapter.

Where do students lose marks in Quadratic Equations?

Not rejecting the negative or impossible root in word problems (negative speed, age, length). Fix: write 'x = −60 is rejected as speed cannot be negative'; the scheme gives a mark for it. Using the formula without writing a, b, c and D first. Fix: list a = , b = , c = , D = b² − 4ac = … before substituting; these are step marks.

Are the Quadratic Equations important questions free?

Yes. The 10 important questions on this page are free to try, from 1-mark MCQs to a case study. 5 worked answers are open on the page; the others open with a free account (no card).

Are these NCERT questions?

No. Every practice question here is original, written in the board's question styles with step marks. For the textbook itself, use the NCERT link on this page.

Work through the Route to 95 with your progress saved

Save your answers and track your progressA free account remembers the questions you have done and brings you back to this page.

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Revision notes

Quadratic Equations — revision notes

1. Standard form

2. Solving by factorisation (splitting the middle term)

3. Quadratic formula

4. Nature of roots

5. Situational problems

Worked example 1

Worked example 2

Worked example 3

Common errors

Board-exam tips

Quadratic Equations Class 10: important questions

Section A style · Multiple choice and assertion–reason (1 mark each)

Section B style · Short answer (2 marks each)

Section C style · Short answer (3 marks each)

Section D style · Long answer (5 marks each)

Section E style · Case study (4 marks)

Route to 95: four steps

Secure the basics

Board standard

Full marks on long answers

95+ stretch (HOTS)

Practice questions

Full worked solution Free, no sign-in

Where marks are lost in Quadratic Equations

Quadratic Equations in our sample papers

Quadratic Equations: common questions

Work through the Route to 95 with your progress saved