Quadratic equations show up everywhere — from Class 10 CBSE boards to JEE Main to SAT Math. They carry heavy weightage, they’re predictable, and with the right approach, they’re free marks. Let’s cover everything you need.
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2. That means the highest power of the variable is 2. The word “quadratic” comes from “quadratus” — Latin for square.
The standard form is:
ax² + bx + c = 0, where a ≠ 0
Here, a, b, and c are real number coefficients, and x is the variable. The condition a ≠ 0 is crucial — if a were 0, we’d just have a linear equation.
Some examples to recognise the standard form:
- x² − 5x + 6 = 0 → a = 1, b = −5, c = 6
- 2x² + 3x − 2 = 0 → a = 2, b = 3, c = −2
- x² − 4 = 0 → a = 1, b = 0, c = −4
Every quadratic equation has exactly two roots (also called zeros or solutions). They may be real and distinct, real and equal, or complex (non-real). This is guaranteed by the Fundamental Theorem of Algebra.
The Discriminant — Know the Nature Before You Solve
Before picking a solving method, check the discriminant. This tells you what kind of roots to expect.
D = b² − 4ac
| Value of D | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots (one repeated root) |
| D < 0 | No real roots (complex conjugate roots) |
For JEE and CBSE, questions on nature of roots use only the discriminant — you don’t need to actually find the roots. This is a scoring shortcut.
Method 1: Factoring (Splitting the Middle Term)
Factoring works cleanly when the roots are rational numbers. It’s the fastest method when it applies.
Why it works: If ax² + bx + c = (px + q)(rx + s), then the roots are x = −q/p and x = −s/r. We’re reverse-engineering the multiplication.
The technique: We need to split the middle term bx into two parts whose coefficients multiply to give ac and add to give b.
Example: Solve x² + 5x + 6 = 0
Step 1: Here a = 1, b = 5, c = 6. Find two numbers that multiply to ac = 6 and add to b = 5. Those numbers are 2 and 3.
Step 2: Rewrite: x² + 2x + 3x + 6 = 0
Step 3: Group and factor: x(x + 2) + 3(x + 2) = 0
Step 4: Take the common factor: (x + 2)(x + 3) = 0
Step 5: So x = −2 or x = −3.
For JEE MCQs, always try factoring first — it takes 15 seconds. If factors don’t come within 10 seconds, switch to the formula. Don’t waste time.
Method 2: Quadratic Formula
This works for every quadratic equation, no exceptions. When factoring fails (especially for irrational or complex roots), this is your go-to method.
x = (−b ± √(b² − 4ac)) / 2a
Example: Solve 2x² − 7x + 3 = 0
Step 1: Identify a = 2, b = −7, c = 3
Step 2: Calculate D = (−7)² − 4(2)(3) = 49 − 24 = 25
Step 3: √D = √25 = 5
Step 4: x = (7 ± 5) / 4
Step 5: x = 12/4 = 3 or x = 2/4 = 1/2
So the roots are x = 3 and x = 1/2.
In CBSE boards, the quadratic formula question is almost guaranteed. Write each step clearly — discriminant calculation, substitution, simplification — because step marks are awarded even for partially correct solutions.
Method 3: Completing the Square
This method is less common for solving, but it’s conceptually important. It’s also how the quadratic formula is derived. Some JEE questions specifically ask you to use this method.
Why it works: We convert ax² + bx + c into the form a(x + h)² + k, which is easy to solve by taking square roots.
Example: Solve x² + 6x + 5 = 0 by completing the square
Step 1: Move the constant: x² + 6x = −5
Step 2: Take half the coefficient of x, square it: (6/2)² = 9. Add to both sides: x² + 6x + 9 = −5 + 9
Step 3: Left side is a perfect square: (x + 3)² = 4
Step 4: Take square root: x + 3 = ±2
Step 5: x = −3 + 2 = −1 or x = −3 − 2 = −5
When a ≠ 1, divide the entire equation by a first before completing the square. Many students forget this and get the wrong answer.
Relationship Between Roots and Coefficients
This is a high-value topic for JEE. If α and β are the two roots of ax² + bx + c = 0:
α + β = −b/a
α × β = c/a
These are called Vieta’s formulas. You can form equations given the sum and product of roots using:
x² − (α + β)x + αβ = 0
Example: If the roots are 3 and −5, form the equation. Sum = 3 + (−5) = −2, Product = 3 × (−5) = −15 Equation: x² − (−2)x + (−15) = 0 → x² + 2x − 15 = 0
Solved Examples: Easy to Hard
Easy: x² − 9 = 0
This is a difference of squares. x² = 9, so x = ±3. Done in one line.
Medium: 3x² + 5x − 2 = 0
Using formula: D = 25 + 24 = 49. x = (−5 ± 7) / 6. So x = 2/6 = 1/3 or x = −12/6 = −2.
Hard: Word Problem — Consecutive integers
The product of two consecutive positive integers is 56. Find them.
Let the integers be n and n + 1. n(n + 1) = 56 n² + n − 56 = 0 D = 1 + 224 = 225, √D = 15 n = (−1 + 15)/2 = 7 or n = (−1 − 15)/2 = −8
Since we need positive integers, n = 7. The integers are 7 and 8.
Exam Tips
For CBSE Class 10:
- The quadratic formula question (3 or 4 marks) is always there. Practice writing it neatly.
- Discriminant-based nature of roots questions come for 1-2 marks. Just calculate D and state the nature.
- Word problems (age, number, geometry) form 5-6 marks. Read carefully and form the correct equation first.
For JEE Main:
- Questions on sum/product of roots are very common.
- Quadratic inequalities are a favourite topic — know the sign scheme.
- “For what value of k does the equation have equal roots?” type questions always use D = 0.
For SAT Math:
- Factor when possible — it’s fastest.
- The discriminant concept appears in “how many solutions” questions.
- Vertex form is often tested: a(x − h)² + k.
5 Common Mistakes
Mistake 1: Forgetting ± in the square root x² = 9 does NOT give x = 3 only. It gives x = ±3. Always write both roots.
Mistake 2: Sign error in Vieta’s formulas Sum of roots = −b/a, NOT b/a. The negative sign is critical. In x² − 5x + 6 = 0, sum = −(−5)/1 = 5, not −5.
Mistake 3: Not checking if the equation is in standard form Always rearrange to ax² + bx + c = 0 before applying any formula. Equations like 3x² = 2x + 1 need rearranging first.
Mistake 4: Dividing by x directly If you have x² = 5x, don’t divide both sides by x. You’ll lose the root x = 0. Factor instead: x(x − 5) = 0.
Mistake 5: Completing the square with a ≠ 1 When a ≠ 1, divide the whole equation by a first. Trying to complete the square with a leading coefficient other than 1 creates messy fractions and errors.
Real-World Examples
Example 1: ISRO’s Vikram Lander and Projectile Height
When ISRO engineers planned the Chandrayaan-3 descent, they modelled the lander’s altitude using equations of the form . At any fixed horizontal slice — “at what time does m?” — you are solving a quadratic in . The two solutions tell you the two moments the craft passes that altitude: once on the way down, once if it overshoots and comes back. Engineers keep only the physically meaningful root.
Connect to the syllabus: This is the discriminant in action — gives two real roots (two crossing times), gives one (just grazes that height), and means the craft never reaches it.
Example 2: Wagh Bakri’s Rectangular Tea Garden
A tea estate in Munnar has 200 metres of fencing to enclose a rectangular plot alongside a river (so only three sides need fencing). The owner wants exactly 4800 m² of growing area. Setting width , the constraint gives , which simplifies to , or . The roots, m and m, are both valid rectangles — same area, different shapes.
Connect to the syllabus: This is the classic area optimisation word problem from CBSE Class 10 Chapter 4; practise converting the English sentence into form before reaching for the quadratic formula.
Example 3: Break-Even for a Mumbai Street-Food Cart
Ravi runs a vada pav stall near Dadar station. His daily cost is rupees (rising steeply at high volumes due to extra staff) and his revenue is . Setting to find break-even: , giving . The roots are and units — below 40 or above 120 vada pavs, Ravi loses money. The profitable window sits between the two roots.
Connect to the syllabus: Recognising that a quadratic inequality holds between the roots (when ) is a recurring JEE Main pattern — the sign-chart method starts exactly here.
Practice Questions
Q1. Solve: x² − 7x + 12 = 0
Factor: find two numbers that multiply to 12, add to −7. Those are −3 and −4. (x − 3)(x − 4) = 0 x = 3 or x = 4
Q2. Find the discriminant of 2x² − 5x + 3 = 0 and state the nature of roots.
D = (−5)² − 4(2)(3) = 25 − 24 = 1 D > 0, so the equation has two distinct real roots.
Q3. Solve using the quadratic formula: x² + x − 6 = 0
a = 1, b = 1, c = −6 D = 1 + 24 = 25 x = (−1 ± 5) / 2 x = 2 or x = −3
Q4. The sum of a number and its reciprocal is 10/3. Find the number.
Let the number be x. Then x + 1/x = 10/3 3x² − 10x + 3 = 0 (3x − 1)(x − 3) = 0 x = 3 or x = 1/3 Both are valid answers.
Q5. For what value of k does kx² + 6x + 1 = 0 have equal roots?
For equal roots, D = 0. D = 36 − 4k = 0 4k = 36 k = 9
Q6. Solve by completing the square: x² − 4x − 5 = 0
x² − 4x = 5 x² − 4x + 4 = 5 + 4 (x − 2)² = 9 x − 2 = ±3 x = 5 or x = −1
Q7. If one root of x² − 3x + k = 0 is 1, find k and the other root.
Substituting x = 1: 1 − 3 + k = 0, so k = 2. Now x² − 3x + 2 = 0 → (x − 1)(x − 2) = 0. The other root is x = 2.
Q8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the original speed.
Let speed = x km/h. Time = 360/x. With increased speed: time = 360/(x + 5). 360/x − 360/(x + 5) = 1 360(x + 5) − 360x = x(x + 5) 1800 = x² + 5x x² + 5x − 1800 = 0 (x + 45)(x − 40) = 0 Since speed must be positive, x = 40 km/h.
Frequently Asked Questions
Can a quadratic equation have only one root? Technically, yes — when D = 0, both roots are equal. We call this a “repeated root” or “double root.” But formally, the equation still has two roots; they just happen to be the same value.
What is the difference between roots and solutions? Nothing — they’re the same thing. “Roots,” “solutions,” and “zeros” all refer to the values of x that satisfy the equation.
Can the roots be fractions? Absolutely. When D is a perfect square, roots are rational (whole numbers or fractions). When D is positive but not a perfect square, roots are irrational (contain surds).
How do I know which method to use? Try factoring first for 10 seconds. If it doesn’t click, use the quadratic formula. Use completing the square only when the question specifically asks for it or when the coefficient of x is even and the equation is monic (a = 1).
Are quadratic equations in JEE just the basics? No. JEE tests quadratics in combination with other topics — inequalities, complex roots, graphs, and conditions for specific root ranges. Build the basics solid first, then move to those.
What happens if a = 0 in ax² + bx + c = 0? It stops being quadratic and becomes linear (bx + c = 0). The definition requires a ≠ 0. If an exam question has a = 0, it’s either a trick or an error.