Quadratic Equations Class 10: Formulas, Tricks & Practice

Q: What is a quadratic equation?

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a is not equal to 0. The condition a ≠ 0 is essential — without it, the equation is no longer degree 2 and stops being quadratic.

Q: What is the quadratic formula?

The quadratic formula gives the roots of ax² + bx + c = 0 directly as x = (−b ± √(b² − 4ac)) / 2a. It works for every quadratic equation, including ones that cannot be factorised easily.

Q: What is the discriminant and what does it tell you?

The discriminant is D = b² − 4ac, the expression under the square root in the quadratic formula. If D is greater than 0, the equation has two distinct real roots. If D equals 0, it has two equal real roots. If D is less than 0, it has no real roots.

Q: What are the three methods to solve a quadratic equation?

The three standard methods are factorisation (splitting the middle term into two parts that multiply to ac and add to b), completing the square (rewriting the equation as a perfect square plus a constant), and the quadratic formula, which works for any quadratic equation regardless of whether it factorises neatly.

Q: Can a quadratic equation have a negative answer that is still valid?

Yes, a negative root can be a mathematically valid solution to the equation. However, in word problems involving real-world quantities like speed, age, length, or the number of items, a negative root is usually rejected because those quantities cannot be negative.

Q: What are the most common mistakes in Class 10 Quadratic Equations?

Common mistakes include forgetting to check a ≠ 0, sign errors while substituting into the quadratic formula, misreading the discriminant condition, not dividing every term by the leading coefficient before completing the square, and accepting a negative root in a word problem where the quantity must be positive.

Quadratic Equations Class 10 formulas tricks and practice questions

Quadratic Equations Class 10: Formulas, Tricks & Practice Questions (2026 Guide)

📅 Updated Sep 4, 2026 ⏱ 13 min read 📘 CBSE / NCERT Class 10, Chapter 4 ✍️ MyTestSeries Expert Team

Quadratic Equations Class 10 — quadratic formula and discriminant explained by MyTestSeries

Quadratic Equations is where the "zeroes of a polynomial" idea from the last chapter turns practical: instead of just relating zeroes to coefficients, you now solve for them directly, using three distinct methods. It's also the chapter that quietly borrows from everything before it — factorisation from Polynomials, the coefficient-based reasoning from Linear Equations, and now a brand-new tool, the discriminant, that predicts what kind of roots you'll get before you even solve.

This guide lays out all three solving methods with a worked example each, the discriminant rule for nature of roots, real exam-style word problems, and the specific mistakes — sign slips, skipped checks, invalid negative answers — that quietly cost marks year after year.

3Solving Methods

6Solved Examples

7Common Mistakes

1Quick Formula Table

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1. What Is a Quadratic Equation?

ax² + bx + c = 0, where a ≠ 0 a, b, c are real numbers · a is the leading coefficient · the condition a ≠ 0 is what makes it degree 2, not degree 1

Any equation that can be rearranged into this exact form is quadratic — regardless of how it's originally written. The values of x that satisfy the equation are called its roots (or solutions), and geometrically they're the x-coordinates of the points where the parabola y = ax² + bx + c crosses the x-axis — the same "zeroes" idea from the Polynomials chapter, now solved for directly.

💡 Quick check: Is (x+2)(x−1) = x² a quadratic equation? Expand the left side: x²+x−2 = x². Simplify: x−2=0. The x² terms cancel, leaving a linear equation — so this is not quadratic, even though it looked like one at first glance.

2. Method 1 — Factorisation

Split the middle term (bx) into two parts whose product equals ac and whose sum equals b. This lets you factor the expression into two linear brackets, each of which is set to zero separately.

If ax² + bx + c = (x − p)(x − q) = 0, then x = p or x = q Fastest method when the equation factorises with small, whole-number roots.

3. Method 2 — Completing the Square

Rewrite the quadratic as a perfect square trinomial plus a constant, then isolate and take the square root of both sides. Unlike factorisation, this method always works — even when the roots aren't whole numbers.

Steps: (1) Divide every term by a so the x² coefficient is 1. (2) Move the constant to the other side. (3) Add (half the x-coefficient)² to both sides. (4) Write the left side as a perfect square and solve.

Three methods, one right answer each time — but only if you practise choosing the fastest one. Try the 7-Day Demo Test Series for instant, explained solutions after every attempt.

4. Method 3 — The Quadratic Formula

x = ( −b ± √(b² − 4ac) ) / 2a Derived by completing the square on the general form ax²+bx+c=0. Works for every quadratic equation, without exception.

This is the universal fallback — when factorisation isn't obvious and completing the square feels slow, the quadratic formula gets you to the answer directly, as long as you substitute a, b, and c carefully.

5. The Discriminant & Nature of Roots

D = b² − 4ac The expression under the square root in the quadratic formula — its sign alone tells you what kind of roots the equation has, before you fully solve it.

Discriminant (D)	Nature of Roots
D > 0	Two distinct real roots
D = 0	Two equal real roots (a repeated root)
D < 0	No real roots

⚠️ Watch out: "D = 0" gives two equal roots, not "one root" — CBSE mark schemes specifically expect you to state that the two roots coincide, not just that there's a single value.

6. Six Fully Solved Examples

Example 1 — Factorisation method

Solve x² − 3x − 4 = 0 by factorisation.

Split the middle term: −4 and +1 multiply to −4 and add to −3.
x² − 4x + x − 4 = 0 → x(x−4) + 1(x−4) = 0 → (x−4)(x+1) = 0
x = 4 or x = −1

Example 2 — Completing the square

Solve 2x² − 7x + 3 = 0 by completing the square.

Divide by 2: x² − (7/2)x + 3/2 = 0 → x² − (7/2)x = −3/2
Add (7/4)² = 49/16 to both sides: x² − (7/2)x + 49/16 = −3/2 + 49/16 = 25/16
(x − 7/4)² = 25/16 → x − 7/4 = ±5/4
x = 3 or x = 1/2

Example 3 — Quadratic formula

Solve 6x² − x − 2 = 0 using the quadratic formula.

a=6, b=−1, c=−2. D = (−1)² − 4(6)(−2) = 1 + 48 = 49.
x = (1 ± √49) / 12 = (1 ± 7) / 12
x = 8/12 = 2/3, or x = −6/12 = −1/2

Example 4 — Nature of roots

Without solving, find the nature of the roots of 2x² − 4x + 3 = 0.

a=2, b=−4, c=3. D = (−4)² − 4(2)(3) = 16 − 24 = −8.
Since D < 0, the equation has no real roots.

Example 5 — Word problem (consecutive integers)

The product of two consecutive positive integers is 306. Find the integers.

Let the integers be x and x+1. Then x(x+1) = 306 → x² + x − 306 = 0.
D = 1 + 1224 = 1225 = 35². x = (−1 ± 35)/2 → x = 17 or x = −18.
Since the integers must be positive, reject x = −18. The integers are 17 and 18.

Example 6 — Word problem (speed and time)

A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Find the original speed.

Let original speed = x km/hr. Original time = 360/x. New time = 360/(x+5), which is 1 hour less:
360/x − 360/(x+5) = 1 → 360(x+5) − 360x = x(x+5) → 1800 = x² + 5x → x² + 5x − 1800 = 0
D = 25 + 7200 = 7225 = 85². x = (−5 ± 85)/2 → x = 40 or x = −45.
Speed cannot be negative, so reject x = −45. Original speed = 40 km/hr.

7. Common Mistakes Students Make

⚠️ Mistake 1 — Not checking a ≠ 0. If the coefficient of x² turns out to be zero after simplification, the equation is linear, not quadratic — always simplify fully before applying any quadratic method.

⚠️ Mistake 2 — Sign errors substituting into the quadratic formula. When b is already negative (e.g., b = −4), −b becomes positive — a very common place to drop a sign.

⚠️ Mistake 3 — Misreading the discriminant condition. Mixing up D ≥ 0 (real roots, possibly equal) with D > 0 (strictly two distinct real roots) is one of the most frequent conceptual slips in this chapter.

⚠️ Mistake 4 — Forgetting to divide every term before completing the square. If a ≠ 1, every single term — including the constant — must be divided by a first, or the "add (half the coefficient)²" step gives the wrong number.

⚠️ Mistake 5 — Accepting an invalid negative root. In word problems about speed, age, length, or count, a negative root is mathematically correct but physically impossible — always state clearly why it's rejected (see Examples 5 and 6).

⚠️ Mistake 6 — Arithmetic slips while expanding brackets. Errors in (x−p)(x−q) expansion during factorisation are common — always re-expand your factored answer mentally to confirm it matches the original equation.

⚠️ Mistake 7 — Not verifying the final roots. Substitute each root back into the original equation. This quick check catches almost every sign or arithmetic error before it costs marks.

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8. Quick Formula Recap Table

Concept	Formula / Condition
Standard form	ax² + bx + c = 0, a ≠ 0
Quadratic formula	x = (−b ± √(b²−4ac)) / 2a
Discriminant	D = b² − 4ac
Two distinct real roots	D > 0
Two equal real roots	D = 0
No real roots	D < 0
Sum of roots	−b/a
Product of roots	c/a

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9. Exam Tricks & How to Revise

⚡ Check the discriminant before you commit to a method. If D isn't a perfect square, factorisation won't give clean roots — switch straight to the quadratic formula and save time.

✍️ Memorise the formula as a rhythm, not a string of symbols. Say it out loud: "minus b, plus or minus root of b-squared minus 4ac, all over 2a" — this prevents mid-exam sign slips.

🧮 For word problems, define your variable in one sentence before writing the equation. Examples 5 and 6 above both start this way — it prevents the classic mistake of solving for the wrong quantity.

🎯 Always test whether a root makes physical sense. Speed, age, length, and count can never be negative — flag and reject invalid roots explicitly, don't just drop them silently.

🔁 Revisit this page before your unit test. Bookmark it, and use the Quick Formula Recap Table (Section 8) as your final five-minute revision before the exam.

📚 Also Useful on MyTestSeries

🌐 Trusted External References

NCERT Official Website — download the official Class 10 Maths textbook
CBSE Academic — official syllabus & sample papers
Quadratic Equation — Wikipedia — broader mathematical background
Discriminant — Wikipedia — extended theory beyond the Class 10 syllabus

10. Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. The condition a ≠ 0 keeps the equation at degree 2.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) / 2a. It gives the roots of any quadratic equation directly, without needing to factorise.

What is the discriminant and what does it tell you?

D = b² − 4ac. D > 0 means two distinct real roots, D = 0 means two equal real roots, and D < 0 means no real roots.

What are the three methods to solve a quadratic equation?

Factorisation, completing the square, and the quadratic formula. The formula always works; factorisation is fastest when roots are simple whole numbers.

Can a quadratic equation have a negative answer that is still valid?

Yes, mathematically. But in word problems involving speed, age, length, or count, a negative root is usually rejected because those real-world quantities can't be negative.

What are the most common mistakes in Class 10 Quadratic Equations?

Not checking a ≠ 0, sign errors in the quadratic formula, misreading the discriminant condition, forgetting to divide every term before completing the square, and accepting an invalid negative root in a word problem.

Conclusion: One Equation, Three Doors In

Quadratic Equations gives you three different ways to reach the same answer — factorisation when the numbers cooperate, completing the square when you need to understand the structure, and the quadratic formula when nothing else is faster. The discriminant, meanwhile, lets you predict the type of answer before you've solved anything at all — a genuinely useful shortcut in a timed exam.

Rework the six solved examples above without checking the answers first, say the quadratic formula out loud until the rhythm is automatic, and always ask whether a negative root makes sense in context. That combination — method, formula, and judgment — is what actually turns this chapter into consistent marks.

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Tags: Quadratic Equations Class 10, Quadratic Formula, Discriminant, Completing the Square, Factorisation Method, Nature of Roots, CBSE 2026, NCERT Class 10 Maths Chapter 4

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