01. What is a Quadratic Equation?
A quadratic equation in one variable is an equation of the form:
where
a, b and c are real numbers and:
a ≠ 0
💡 Easy Hindi:
Quadratic equation mein variable ki highest power 2 hoti hai.
Agar x² present hai aur uska coefficient zero nahi hai,
toh equation quadratic ho sakti hai.
✏ Example
2x² + 5x − 3 = 0
Here:
a = 2
b = 5
c = −3
Since a ≠ 0, it is a quadratic equation.
02. How to Identify a Quadratic Equation
A quadratic equation must have:
✓ Highest power of variable = 2
✓ Coefficient of x² must not be zero
✓ It can be written in the form:
ax² + bx + c = 0
✏ Example 1
3x² − 7x + 2 = 0
Quadratic ✓
✏ Example 2
5x + 8 = 0
Not quadratic ✗
Because highest power is 1.
💡 Hindi:
Power 1 → Linear
Power 2 → Quadratic
Power 3 → Cubic
03. Converting into Standard Form
Before solving a quadratic equation, arrange it in:
ax² + bx + c = 0
✏ Example
Solve:
x² + 5 = 3x
Bring all terms to one side:
x² − 3x + 5 = 0
Therefore:
a = 1, b = −3, c = 5
💡 Pehle equation ko zero ke equal karo.
Uske baad a, b, c identify karo.
04. Roots / Solutions of a Quadratic Equation
The values of the variable which satisfy the quadratic equation
are called its roots or solutions.
✏ Example
Consider:
x² − 5x + 6 = 0
Factorise:
(x − 2)(x − 3) = 0
Therefore:
x = 2 or x = 3
Roots are 2 and 3.
💡 Root ka matlab: woh value jo equation mein put karne par
equation ko true bana de.
05. Factorisation Method ⭐⭐⭐
A quadratic equation can sometimes be solved by factorising it
into two linear factors.
Basic idea:
ax² + bx + c = 0
→ Factorise
→ Product of two factors = 0
→ Use zero product property
✏ Example
Solve:
x² − 7x + 12 = 0
We need two numbers whose:
Product = 12
Sum = −7
Numbers are −3 and −4.
Therefore:
x² − 7x + 12 = 0
(x − 3)(x − 4) = 0
So:
x − 3 = 0 → x = 3
x − 4 = 0 → x = 4
Roots = 3, 4
💡 Hindi trick:
Middle term ko aise split karo ki product first × last
term ke equal aaye.
06. Factorisation — Splitting the Middle Term
✏ Example
Solve:
2x² + 7x + 3 = 0
Product:
2 × 3 = 6
We need two numbers whose product is 6 and sum is 7:
6 and 1
Split 7x:
2x² + 6x + x + 3 = 0
Group:
2x(x + 3) + 1(x + 3) = 0
(x + 3)(2x + 1) = 0
Therefore:
x + 3 = 0 → x = −3
2x + 1 = 0 → x = −1/2
Roots = −3, −1/2
💡 Jab x² ka coefficient 1 na ho, tab
first coefficient × constant wala product useful hota hai.
07. Zero Product Property
If:
AB = 0
then:
A = 0 or B = 0
✏ Example
(x − 5)(x + 2) = 0
Therefore:
x − 5 = 0
or
x + 2 = 0
Hence:
x = 5 or x = −2
💡 Dono factors ko zero ke equal karke roots nikaalte hain.
08. Quadratic Formula ⭐⭐⭐
For:
ax² + bx + c = 0
identify:
a = coefficient of x²
b = coefficient of x
c = constant term
✏ Example
Solve:
x² − 5x + 6 = 0
Here:
a = 1, b = −5, c = 6
Using formula:
x =
−(−5) ± √[(-5)² − 4(1)(6)]
2(1)
x =
5 ± √(25 − 24)
2
x =
5 ± 1
2
Therefore:
x = 3 or x = 2
Roots = 2, 3
💡 Hindi:
Formula mein values substitute karte waqt signs carefully dekho,
especially b negative ho toh −b positive ho jayega.
09. Quadratic Formula — Fraction Type
✏ Example
Solve:
2x² − 3x − 2 = 0
Here:
a = 2
b = −3
c = −2
x =
−(−3) ± √[(-3)² − 4(2)(−2)]
4
x =
3 ± √(9 + 16)
4
x =
3 ± 5
4
Therefore:
x = 2
or
x = −1/2
Roots = 2, −1/2
10. Discriminant ⭐⭐⭐
The expression:
D = b² − 4ac
is called the
discriminant of a quadratic equation.
💡 Easy Hindi:
Discriminant ko ek "root detector" ki tarah socho.
Ye batata hai ki equation ke real roots ka nature kya hoga.
| Discriminant |
Nature of Roots |
| D > 0 |
Two distinct real roots |
| D = 0 |
Two equal real roots |
| D < 0 |
No real roots |
11. Nature of Roots — D > 0
✏ Example
x² − 5x + 6 = 0
a = 1, b = −5, c = 6
D = b² − 4ac
D = (−5)² − 4(1)(6)
D = 25 − 24
D = 1
Since:
D > 0
The equation has two distinct real roots.
12. Nature of Roots — D = 0
✏ Example
x² − 6x + 9 = 0
D = (−6)² − 4(1)(9)
D = 36 − 36
D = 0
Therefore:
The equation has two equal real roots.
💡 D = 0 means dono roots same hain.
13. Nature of Roots — D < 0
✏ Example
x² + 2x + 5 = 0
D = (2)² − 4(1)(5)
D = 4 − 20
D = −16
Since D < 0:
The equation has no real roots.
💡 CBSE Class 10 mein focus real roots par hai.
Negative discriminant → no real roots.
14. Finding k Using Discriminant ⭐⭐⭐
✏ Example
Find the value of k for which:
x² + 4x + k = 0
has equal roots.
For equal roots:
D = 0
Therefore:
b² − 4ac = 0
Here:
a = 1, b = 4, c = k
16 − 4k = 0
4k = 16
k = 4
Therefore, k = 4.
💡 Equal roots dikhe → immediately D = 0 use karo.
15. Situational Problems — General Method
Step 1: Read the statement carefully.
Step 2: Let the unknown quantity be x.
Step 3: Translate the condition into an equation.
Step 4: Bring the equation to standard form.
Step 5: Solve by factorisation or quadratic formula.
Step 6: Reject any value which is not meaningful in the
given situation.
Step 7: Write the final answer with units/context.
💡 Hindi:
Word problem mein sabse pehla kaam calculation nahi,
equation banana hai.
16. Situational Problem — Consecutive Integers ⭐⭐⭐
✏ Example
The product of two consecutive positive integers is 72.
Find the integers.
Let the first integer = x.
Next integer = x + 1.
According to question:
x(x + 1) = 72
x² + x − 72 = 0
Factorise:
(x + 9)(x − 8) = 0
So:
x = −9 or x = 8
Since integers are positive:
x = 8
Next integer = 9
Therefore, the integers are 8 and 9.
💡 Negative answer mathematically possible ho sakta hai,
but question mein positive integers hain, isliye −9 reject hoga.
17. Situational Problem — Area
✏ Example
The area of a rectangular garden is 96 m².
Its length is 4 m more than its breadth.
Find its dimensions.
Let breadth = x m.
Length = x + 4 m.
Area:
x(x + 4) = 96
x² + 4x − 96 = 0
Factorise:
(x + 12)(x − 8) = 0
x = −12 or x = 8
Length = 8 + 4 = 12 m
Negative dimension is not possible.
Breadth = 8 m, Length = 12 m
18. Situational Problem — Product of Numbers
✏ Example
The product of two positive numbers is 180 and one number
is 3 more than the other. Find the numbers.
Let smaller number = x.
Larger number = x + 3.
x(x + 3) = 180
x² + 3x − 180 = 0
Factorise:
(x + 15)(x − 12) = 0
x = −15 or x = 12
Since numbers are positive:
x = 12
x + 3 = 15
Numbers are 12 and 15.
19. Important Exam Type — Identify a,b,c
✏ Example
For:
3x² − 7x + 2 = 0
Compare with:
ax² + bx + c = 0
Therefore:
a = 3
b = −7
c = 2
💡 Sign mat bhoolna!
b = −7 hai, +7 nahi.
20. Important Exam Type — Is it Quadratic?
✏ Example
Check whether:
(x + 2)(x + 1) = x² + 3
is a quadratic equation.
Expand LHS:
x² + 3x + 2 = x² + 3
Cancel x²:
3x + 2 = 3
3x − 1 = 0
This is a linear equation.
Therefore, it is NOT a quadratic equation.
21. Roots and Verification
✏ Example
For:
x² − 5x + 6 = 0
Suppose x = 2.
Put x = 2:
(2)² − 5(2) + 6
= 4 − 10 + 6
= 0
Therefore, x = 2 is a root.
💡 Root verify karne ke liye value ko original equation mein
substitute karo.
22. Important Relationship with Roots
If α and β are the roots of:
ax² + bx + c = 0
then:
α + β = −b/a
and:
αβ = c/a
✏ Example
For:
2x² − 7x + 3 = 0
Here:
a = 2, b = −7, c = 3
Sum of roots:
α + β = −(−7)/2
= 7/2
Product of roots:
αβ = 3/2
💡 Ye relationship roots ko directly calculate kiye bina
unka sum aur product find karne mein useful hai.
23. Forming a Quadratic Equation from Roots
If α and β are the roots, then the quadratic equation is:
x² − (α + β)x + αβ = 0
✏ Example
Form a quadratic equation whose roots are 2 and 5.
α + β = 7
αβ = 10
Therefore:
x² − 7x + 10 = 0
Required equation: x² − 7x + 10 = 0
24. Common Mistakes ⚠️
1. Equation ko standard form mein convert kiye bina
a, b, c identify mat karo.
2. b negative ho toh −b carefully handle karo.
3. Quadratic formula mein denominator
2a hota hai.
4. √ ke andar calculation mein sign mistake mat karo.
5. ± ko ignore mat karo — generally two values milengi.
6. Word problems mein negative length, age or distance
ko automatically accept mat karo.
7. Discriminant ka formula:
D = b² − 4ac
exactly yaad rakho.
25. One-Page Formula Revision ⭐⭐⭐
| Condition |
Nature of Roots |
| D > 0 |
Two distinct real roots |
| D = 0 |
Two equal real roots |
| D < 0 |
No real roots |
26. THE GOAL — Exam Strategy 🎯
Question says “solve by factorisation”?
→ Factorise → zero product property → roots.
Question says “solve using formula”?
→ Identify a, b, c → substitute in formula → simplify.
Question asks “nature of roots”?
→ Calculate D = b² − 4ac.
Question says “equal roots”?
→ Put D = 0.
Question gives a real-life situation?
→ Define variable → form quadratic equation → solve →
interpret the answer.
💡 Final Hindi Tip:
Is chapter ka main idea sirf formula yaad karna nahi hai.
Pehle ye samjho ki question mein unknown kya hai,
equation kaise ban rahi hai, aur phir decide karo
factorisation use karni hai ya quadratic formula.