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

Quadratic Equations Class 10: notes and important questions

Revision notes, 39 board-style questions with the step marks shown, and a four-step route from the basics to 95+, each step ending in a short checkpoint.

  • 39 questions
  • 12 multiple choice, 3 assertion–reason, 8 very short answer, 8 short answer, 5 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).

Revision notes

Quadratic Equations — revision notes

1. Standard form

\(ax^2 + bx + c = 0\) 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 \(ac\) and whose sum is \(b\). E.g. \(6x^2 + 7x - 20\): \(ac = -120 = 15 \times (-8)\), so \(6x^2 + 15x - 8x - 20 = (2x + 5)(3x - 4)\); roots \(-\tfrac52, \tfrac43\).

3. Quadratic formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \qquad D = b^2 - 4ac\]

4. Nature of roots

  • \(D \gt 0\): two distinct real roots.
  • \(D = 0\): two equal real roots, each \(-\dfrac{b}{2a}\).
  • \(D \lt 0\): no real roots.

5. Situational problems

  • Speed–time: time \(= \dfrac{\text{distance}}{\text{speed}}\); form “time difference” equations.
  • Work/taps: rates add: \(\dfrac1x + \dfrac1y = \dfrac1T\).
  • Always reject roots that are negative or otherwise impossible, and say why.

Worked example 1

Solve \(x^2 + 2x - 4 = 0\): \(D = 4 + 16 = 20\), \(x = \dfrac{-2 \pm 2\sqrt5}{2} = -1 \pm \sqrt5\).

Worked example 2

Find \(k\) so that \(x^2 - kx + 16 = 0\) 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. \(\dfrac{300}{x} - \dfrac{300}{x+10} = 1 \Rightarrow x^2 + 10x - 3000 = 0 \Rightarrow (x + 60)(x - 50) = 0\), 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 \(2a\).
  • 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.

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. 9 of the 39 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?

  1. (a)\((x + 2)^2 = x^2 + 5\)
  2. (b)\(x(x + 1) + 8 = (x + 2)(x - 2)\)
  3. (c)\((x - 3)(2x + 1) = x(x + 5)\)
  4. (d)\(x + 3 = 2x - 7\)
Q2·1 mark·Multiple choiceStandard form of a quadratic equation

If \(x = -2\) is a root of \(3x^2 + 7x + p = 0\), then \(p\) equals

  1. (a)\(26\)
  2. (b)\(-2\)
  3. (c)\(2\)
  4. (d)\(-26\)
Q3·1 mark·Multiple choiceSolution by factorisation

The roots of \(x^2 - 7x + 10 = 0\) are

  1. (a)\(1, 10\)
  2. (b)\(-2, -5\)
  3. (c)\(2, 5\)
  4. (d)\(-2, 5\)

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.

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).