Quadratic Equations - Quick Notes & Formula Sheet
Class 10 CBSE Mathematics · Chapter 4 · Complete Revision Guide (2026-27)
Syllabus Note for 2026-27
No topics have been officially deleted from the Quadratic Equations chapter this year. The one change worth knowing: Completing the Square is no longer a standalone examinable exercise - but it's still worth understanding, since it's how the quadratic formula itself is derived. Factorisation, the quadratic formula, and the discriminant/nature of roots remain fully examinable and high-weightage.
1. Standard Form of a Quadratic Equation
ax² + bx + c = 0, where a ≠ 0
Here a, b, and c are real numbers, and x is the variable. The values of x that satisfy this equation are called the roots of the equation.
Trap: Not rearranging to standard form first
A very common error is misidentifying a, b, and c when the equation isn't yet in the form ax²+bx+c=0. Always move every term to one side and simplify BEFORE reading off the coefficients - otherwise sign errors creep into every later step.
2. Methods of Solving a Quadratic Equation
A. Factorisation Method
Split the middle term (bx) into two terms whose coefficients multiply to give a×c and add to give b. Factor by grouping, then set each factor to zero.
Split the middle term (bx) into two terms whose coefficients multiply to give a×c and add to give b. Factor by grouping, then set each factor to zero.
B. Quadratic Formula
x = [ -b ± √(b² - 4ac) ] / 2a
Works for every quadratic equation, including ones that don't factor neatly with integer roots.
C. Completing the Square
Rewrites the equation into the form (x+k)²=n, letting you extract the roots directly. Not a separately tested method in 2026-27, but it's the derivation behind the quadratic formula above.
Rewrites the equation into the form (x+k)²=n, letting you extract the roots directly. Not a separately tested method in 2026-27, but it's the derivation behind the quadratic formula above.
Quick check tip
Try factorisation first (it's faster when it works). If the numbers don't factor cleanly, switch to the quadratic formula rather than spending time forcing a factorisation that doesn't exist with integers.
3. Discriminant & Nature of Roots (Highest Weightage Topic)
D = b² - 4ac
| Condition | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots (repeated root) |
| D < 0 | No real roots |
Trap: D=0 vs D≥0 confusion
This is the single most common mistake in this chapter. If a question asks for equal roots, you need D=0 exactly. If it asks for real roots, you need D≥0 (which includes both the D>0 and D=0 cases). Students very often apply the wrong one - always re-read the exact wording ("equal" vs "real") before setting up your condition.
4. Relationship Between Roots and Coefficients (Vieta's Formulas)
If α and β are the roots of ax²+bx+c=0:
Sum of roots: α + β = -b/a
Product of roots: α × β = c/a
These let you answer questions about the roots (like α²+β², or forming a new equation from transformed roots) without ever solving for the individual root values.
5. Word Problem Patterns (Very High Exam Weightage)
| Problem Type | Key Setup |
|---|---|
| Consecutive numbers | Represent as x and (x+1), or x and (x+2) for consecutive odd/even numbers |
| Speed-distance-time | Time = Distance/Speed; if speed changes by a fixed amount, represent the new speed as (x±k) |
| Area / geometry (rectangular plot, etc.) | Represent one dimension in terms of the other using the given relationship, then apply Area = length × breadth |
| Ages | Represent present ages with one variable; carefully track "years ago" vs "years hence" as subtraction/addition |
| Work and time | If a task takes (x) days for one worker and (x+k) for another, use combined work-rate equations: 1/x + 1/(x+k) = 1/(time together) |
Trap: Forgetting to reject invalid roots
Quadratic word problems often produce two mathematically valid roots, but only one makes sense in context (e.g., a negative value for time, speed, or length is impossible). Always check both roots against the real-world constraints of the problem and reject the invalid one - this is a scored step in CBSE's marking scheme, not optional.
6. Full Recap - Every Trap in One Place
| Trap | Correct Understanding |
|---|---|
| Reading off a, b, c before rearranging to standard form | Always simplify to ax²+bx+c=0 first |
| Using D=0 when the question asks for "real roots" | "Real roots" needs D≥0; "equal roots" needs D=0 exactly |
| Forgetting the quadratic formula's ± sign | Always gives two roots (unless D=0, where both are equal) |
| Not rejecting the invalid root in word problems | Always check both roots against real-world constraints (no negative time, speed, length, etc.) |
| Treating "quadratic polynomial" and "quadratic equation" as identical | A polynomial has zeroes; setting it equal to 0 makes it an equation with roots |
| Spending exam time deeply practicing Completing the Square as a standalone method | Not separately tested in 2026-27 - know it conceptually, but prioritise factorisation and the formula |
Frequently Asked Questions
What is the discriminant of a quadratic equation?
The discriminant is D = b² - 4ac for a quadratic equation ax²+bx+c=0. It tells you the nature of the roots without needing to solve the equation.
What is the difference between a quadratic equation and a quadratic polynomial?
A quadratic polynomial is the expression ax²+bx+c on its own. A quadratic equation sets that expression equal to zero. The polynomial has zeroes; the equation has roots - closely related terms, but not interchangeable in exam answers.
Is Completing the Square still tested in the CBSE 2026-27 board exam?
It's not a separate standalone exercise in the current NCERT syllabus, but it remains conceptually important since it's used to derive the quadratic formula. Don't skip understanding it, but don't over-invest exam-prep time on it either.
What is the most common mistake students make with the discriminant condition?
Confusing "equal roots" (D=0 exactly) with "real roots" (D≥0, which includes both D>0 and D=0). Always check the exact wording of the question first.
Practice What You've Just Revised
This chapter now has an expanded 5-part quiz series (100 questions total) - a dedicated Discriminant/Nature of Roots part has been added given how heavily this concept is tested.
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