Class 10 Quadratic Equations Notes: Methods, Formula and Board Questions

Table of Contents

1. Why Quadratic Equations Deserves Serious Time
2. What a Quadratic Equation Is
3. Method 1: Factorisation
4. Method 2: The Quadratic Formula and Nature of Roots
5. Word Problems, and the Mistakes That Cost Marks
6. FAQs

Quadratic Equations is one of the most reliable scoring chapters in Class 10 maths. It is short, it is mostly method, and it is asked in a small number of predictable ways. A student who practises it properly can expect to collect nearly every mark it carries.

It is also part of Algebra, the highest-weighted unit in the paper at 20 of the 80 theory marks. And it feeds other chapters. Arithmetic Progressions, heights and distances, and surface area problems often end in a quadratic equation you have to solve.

These notes cover the full chapter in the order the board tests it, with worked examples and the mistakes that lose marks. For the complete chapter notes with a question bank, see ZetaGalaxy Class 10 notes.

Why Quadratic Equations Deserves Serious Time

ZetaGalaxy mapped past CBSE Class 10 maths board papers question by question. In that analysis, Quadratic Equations ranked fifth among all fourteen chapters. Within it, one topic stood out. Solving a quadratic equation by factorisation was asked in 185 questions across the papers analysed, carrying 612 marks in total. That puts it among the ten most-asked topics in the entire Class 10 maths paper.

The pattern is clear. Factorisation is the skill the board tests most in this chapter. The quadratic formula and the nature of roots come next, often as 1-mark or 2-mark questions. Word problems that lead to a quadratic equation appear as longer questions and case studies.

So prepare in that order. Make factorisation fast and accurate, then the formula and discriminant, then word problems.

What a Quadratic Equation Is

A quadratic equation in x is an equation of the form ax² + bx + c = 0, where a, b and c are real numbers and a is not zero. This is called the standard form.

The condition a ≠ 0 matters. If a were zero, the x² term would vanish and the equation would be linear, not quadratic. A common 1-mark question asks for the value of k that makes an equation quadratic, and the answer depends on this condition.

To check whether an equation is quadratic, first simplify it and bring every term to one side. Some equations look quadratic and are not. For example, (x + 2)² = x² + 3 expands to x² + 4x + 4 = x² + 3. The x² terms cancel, leaving 4x + 1 = 0. That is linear.

A root of the equation is a value of x that makes the left side equal zero. A quadratic equation has at most two roots. If α is a root, then aα² + bα + c = 0.

Always write the equation in standard form before choosing a method. Most errors in this chapter begin with skipping this step.

Method 1: Factorisation

Factorisation is the most-asked method, and it is fast once practised. The idea is to split the middle term so the expression breaks into two factors.

For ax² + bx + c, find two numbers whose product is a × c and whose sum is b. Use them to split bx, group the terms, and take out common factors.

Take x² − 5x + 6 = 0. Here a × c = 6 and b = −5. The numbers −2 and −3 multiply to 6 and add to −5. So x² − 2x − 3x + 6 = 0, which gives x(x − 2) − 3(x − 2) = 0, and so (x − 2)(x − 3) = 0. The roots are x = 2 and x = 3.

Now one with a leading coefficient that is not 1. Take 2x² − 7x + 3 = 0. Here a × c = 6 and b = −7. The numbers −6 and −1 work. So 2x² − 6x − x + 3 = 0, which gives 2x(x − 3) − 1(x − 3) = 0, and so (2x − 1)(x − 3) = 0. The roots are x = 1/2 and x = 3.

Check your roots by substituting them back. For x = 3 in the second example, 2(9) − 21 + 3 = 0. It works.

When factorisation is quick, use it. When you cannot find the pair of numbers within a minute, switch to the quadratic formula rather than guessing.

A few patterns make the search faster:

• If c is positive, the two numbers have the same sign as b. If c is negative, they have opposite signs.
• If c is negative, the larger number takes the sign of b.
• Start with the factor pairs of a × c closest to its square root, then move outward.

Take 6x² + x − 2 = 0. Here a × c = −12 and b = 1. Because a × c is negative, the numbers have opposite signs, and the larger one is positive. 4 and −3 work. So 6x² + 4x − 3x − 2 = 0, which gives 2x(3x + 2) − 1(3x + 2) = 0, and so (2x − 1)(3x + 2) = 0. The roots are x = 1/2 and x = −2/3.

Practise until splitting the middle term takes seconds rather than minutes. On a 1-mark or 2-mark question, speed here buys time for the longer questions later in the paper.

Method 2: The Quadratic Formula and Nature of Roots

The quadratic formula solves any quadratic equation in standard form:

x = [−b ± √(b² − 4ac)] / 2a

The expression under the square root, b² − 4ac, is called the discriminant, D. Write D separately first. It keeps the working clean and earns a step mark.

Solve 2x² − 7x + 3 = 0 again with the formula. a = 2, b = −7, c = 3. D = 49 − 24 = 25. So x = (7 ± 5) / 4, which gives x = 3 or x = 1/2. The same roots as before.

The discriminant also tells you the nature of the roots without solving:

• If D > 0, the equation has two distinct real roots.
• If D = 0, it has two equal real roots, each equal to −b/2a.
• If D < 0, it has no real roots.

This is a favourite board question. Find k so that kx² + 6x + 1 = 0 has equal roots. For equal roots, D = 0. So 36 − 4k = 0, which gives k = 9.

Another common type asks you to state the nature of the roots of a given equation. Calculate D, compare it with zero, and write the conclusion in words. Stopping at the number without stating the nature loses the mark.

Word Problems, and the Mistakes That Cost Marks

Word problems turn a situation into a quadratic equation. They follow four steps:

1. Let the unknown be x, and say clearly what x stands for.
2. Write the equation from the conditions in the question.
3. Bring it to standard form and solve.
4. Reject any root that does not make sense, such as a negative length, age or speed, and write the final answer in words.

Try this. The product of two consecutive positive integers is 306. Find them. Let the integers be x and x + 1. Then x(x + 1) = 306, so x² + x − 306 = 0. Split the middle term with 18 and −17, which gives (x + 18)(x − 17) = 0. So x = −18 or x = 17. The integers must be positive, so reject −18. The integers are 17 and 18.

Here is a second type, built on area. A rectangular park is 3 metres longer than it is wide, and its area is 180 square metres. Find its dimensions. Let the width be x metres, so the length is x + 3. Then x(x + 3) = 180, which gives x² + 3x − 180 = 0. The numbers 15 and −12 multiply to −180 and add to 3, so (x + 15)(x − 12) = 0. A width cannot be negative, so x = 12. The park is 12 metres wide and 15 metres long.

The board uses a handful of settings again and again. Ages now and in the past, the speed of a train or boat, the dimensions of a rectangular field or garden, the number of items bought at a price, and consecutive integers. The case-study form of the current paper often wraps one of these in a short story.

The mistakes that cost marks in this chapter are predictable:

• Not writing the equation in standard form before applying a method.
• Sign errors when splitting the middle term or substituting into the formula, especially when b is negative.
• Writing D but not stating the nature of the roots.
• Keeping a root that makes no sense in a word problem, such as a negative age.
• Dividing both sides by x and losing the root x = 0.
• Forgetting that a cannot be zero in a quadratic equation.

A quick checklist before the exam. Can you factorise a quadratic with a ≠ 1 by splitting the middle term? Can you apply the formula without sign errors? Can you decide the nature of the roots from D and find k for equal roots? Can you turn a word problem into an equation and reject the invalid root? If yes, this chapter's marks are within reach.

ZetaGalaxy's chapter notes for Quadratic Equations include practice by marks with answers and board-pattern questions. When a factorisation will not split or a word problem will not form, Nova, the AI teacher, walks through it step by step at your level.

Conclusion

Quadratic Equations rewards practice more than memory. Factorisation carries most of its board marks, the formula and discriminant cover the rest, and word problems follow one clear routine. Practise each until it is automatic, and check every root.

Download the complete chapter notes at ZetaGalaxy Class 10 notes and work through the practice questions.

FREQUENTLY ASKED QUESTIONS

Q: What is the most important topic in Class 10 Quadratic Equations?

A: Solving by factorisation. In ZetaGalaxy's analysis of past board papers, it was asked in 185 questions carrying 612 marks, making it one of the ten most-asked topics in Class 10 maths.

Q: What is the quadratic formula?

A: For ax² + bx + c = 0 with a ≠ 0, x = [−b ± √(b² − 4ac)] / 2a. Write the discriminant b² − 4ac separately before substituting.

Q: How do I find the nature of roots?

A: Calculate D = b² − 4ac. If D > 0, two distinct real roots. If D = 0, two equal real roots. If D < 0, no real roots. Always state the conclusion in words.

Q: When should I use factorisation and when the formula?

A: Use factorisation when you can quickly find two numbers with product a × c and sum b. If you cannot find them within a minute, switch to the quadratic formula, which always works.

Q: Why do I lose marks in quadratic word problems?

A: Usually by not defining x, keeping a root that makes no sense such as a negative age, or not writing the final answer in words. Reject invalid roots and state the answer clearly.