Quadratic Equations – Quadratic Formula
The Quadratic Formula is one of the most important methods for solving quadratic equations in Class 10 Maths. When factorization is difficult or when the factors are not easy to identify, the quadratic formula provides a direct method to find the roots of the equation.
In this chapter, we will learn the standard form of a quadratic equation, how to identify a, b and c, how to apply the quadratic formula, how to calculate the discriminant, how to determine the nature of roots and how to use the relationship between roots and coefficients.
1. 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.
- a ≠ 0.
- The highest power of x is 2.
Example
Consider:
2x² + 5x − 3 = 0
Comparing it with:
ax² + bx + c = 0
we get:
a = 2
b = 5
c = −3
Correctly identifying these three values is the first and one of the most important steps when using the quadratic formula.
2. Why Do We Need the Quadratic Formula?
Quadratic equations can sometimes be solved easily by factorization. However, factorization may become difficult when suitable factors are not easy to find.
For any quadratic equation:
ax² + bx + c = 0
we can directly find its roots using the Quadratic Formula.
Quadratic Formula
x = [−b ± √(b² − 4ac)] / 2a
The symbol ± means that we have to consider both the positive and negative possibilities.
Therefore, the formula gives two possible roots:
x₁ = [−b + √(b² − 4ac)] / 2a
x₂ = [−b − √(b² − 4ac)] / 2a
Formula Memory Trick
Remember the formula in this order:
−b → ± → √(b² − 4ac) → 2a
Do not forget the ± sign because it is responsible for finding both roots.
3. Identify a, b and c Correctly
Before applying the quadratic formula, always compare the equation with:
ax² + bx + c = 0
Then identify the values of a, b and c.
Example 1
2x² − 7x + 3 = 0
Therefore:
a = 2
b = −7
c = 3
Example 2
x² − 5x − 6 = 0
Therefore:
a = 1
b = −5
c = −6
Important: The signs of b and c must be copied exactly from the equation.
A wrong sign can change the value of the discriminant and ultimately give completely incorrect roots.
4. Step-by-Step Method to Solve a Quadratic Equation
Let's understand the complete method using an example.
Example: Solve 2x² − 7x + 3 = 0
Step 1: Identify a, b and c
Comparing:
2x² − 7x + 3 = 0
with:
ax² + bx + c = 0
we get:
a = 2
b = −7
c = 3
Step 2: Calculate b² − 4ac
b² − 4ac = (−7)² − 4(2)(3)
= 49 − 24
= 25
Step 3: Apply the Quadratic Formula
Formula:
x = [−b ± √(b² − 4ac)] / 2a
Substituting the values:
x = [7 ± √25] / 4
Therefore:
x = [7 ± 5] / 4
Step 4: Find Both Roots
First root:
x₁ = (7 + 5) / 4
x₁ = 12 / 4 = 3
Second root:
x₂ = (7 − 5) / 4
x₂ = 2 / 4 = 1/2
Final Answer
x = 3, 1/2
5. What is the Discriminant?
The expression:
b² − 4ac
is called the Discriminant.
It is represented by the symbol:
D = b² − 4ac
The discriminant is extremely important because it tells us the nature of the roots before we actually calculate them.
Three Possible Cases
| Discriminant | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Two equal real roots |
| D < 0 | No real roots |
Quick Trick
Positive → 2 different roots
Zero → Same roots
Negative → No real roots
6. Nature of Roots
The discriminant allows us to determine the nature of the roots without necessarily solving the entire equation.
Case 1: D > 0
If:
D > 0
then the equation has:
Two distinct real roots
Example
x² − 5x + 6 = 0
Here:
a = 1, b = −5, c = 6
Therefore:
D = (−5)² − 4(1)(6)
= 25 − 24 = 1
Since:
D > 0
the equation has two distinct real roots.
Case 2: D = 0
If:
D = 0
then the equation has:
Two equal real roots
In this situation, both roots have the same value.
Case 3: D < 0
If:
D < 0
then the equation has:
No real roots
This is because the square root of a negative number does not give a real number.
7. Equal Roots
When:
D = 0
the quadratic equation has equal roots.
Example: x² − 6x + 9 = 0
Here:
a = 1, b = −6, c = 9
Calculate the discriminant:
D = (−6)² − 4(1)(9)
= 36 − 36
= 0
Therefore, the equation has equal roots.
For equal roots:
x = −b / 2a
Substituting:
x = −(−6) / 2(1)
x = 6 / 2 = 3
Therefore:
x₁ = x₂ = 3
8. Relationship Between Roots and Coefficients
If the roots of:
ax² + bx + c = 0
are α and β, then there are two important relationships.
Sum of Roots
α + β = −b/a
Product of Roots
αβ = c/a
These relationships are very useful for checking answers and solving certain board-level questions.
Example
Consider:
2x² − 7x + 3 = 0
Here:
a = 2, b = −7, c = 3
Therefore:
α + β = −(−7)/2 = 7/2
and:
αβ = 3/2
Our roots were:
3 and 1/2
Check the sum:
3 + 1/2 = 7/2 ✓
Check the product:
3 × 1/2 = 3/2 ✓
Therefore, the roots are verified.
9. Board-Level Example 1
Solve: x² − 7x + 12 = 0
Comparing with:
ax² + bx + c = 0
we get:
a = 1, b = −7, c = 12
Step 1: Find D
D = (−7)² − 4(1)(12)
D = 49 − 48 = 1
Step 2: Apply Formula
x = [7 ± √1] / 2
x = [7 ± 1] / 2
Step 3: Find Roots
x₁ = (7 + 1)/2 = 8/2 = 4
x₂ = (7 − 1)/2 = 6/2 = 3
Answer: x = 4, 3
10. Board-Level Example 2
Solve: 3x² − 2x − 1 = 0
Here:
a = 3, b = −2, c = −1
Step 1: Calculate D
D = (−2)² − 4(3)(−1)
D = 4 + 12 = 16
Step 2: Apply Formula
x = [2 ± √16] / 6
x = [2 ± 4] / 6
Step 3: Find Roots
x₁ = (2 + 4)/6 = 6/6 = 1
x₂ = (2 − 4)/6 = −2/6 = −1/3
Answer: x = 1, −1/3
11. Board-Level Example 3 – Equal Roots
Find k if x² + kx + 9 = 0 has equal roots.
For equal roots:
D = 0
Here:
a = 1, b = k, c = 9
Therefore:
D = k² − 4(1)(9)
k² − 36 = 0
Therefore:
k² = 36
Taking the square root:
k = ±6
Hence:
k = 6 or −6
12. Common Mistakes Students Make
The quadratic formula is easy to use, but small calculation mistakes can change the complete answer.
- ❌ Forgetting to write the equation in standard form.
- ❌ Taking the wrong value of b.
- ❌ Missing a negative sign.
- ❌ Writing the denominator 2a incorrectly.
- ❌ Forgetting the ± sign.
- ❌ Calculating b² incorrectly.
- ❌ Finding only one root.
- ❌ Not checking the answer.
Golden Rule
Standard Form → a, b, c → D → Formula → 2 Roots → Check
13. How to Avoid Sign Mistakes?
Sign mistakes are one of the most common problems when applying the quadratic formula.
For example:
2x² − 7x + 3 = 0
Do not write:
❌ b = 7
The correct value is:
✅ b = −7
When calculating b²:
(−7)² = 49
Remember that the negative sign is included inside the square.
Similarly, if:
c = −3
then write:
4ac = 4(a)(−3)
Do not accidentally change it to positive.
Best Practice
Always write:
a = ___
b = ___
c = ___
before calculating the discriminant.
14. Quadratic Formula – Complete Solving Flow
Quadratic Equation
↓
Write in Standard Form
↓
ax² + bx + c = 0
↓
Find a, b, c
↓
Calculate D = b² − 4ac
↓
Check Nature of Roots
↓
Apply x = [−b ± √D] / 2a
↓
Calculate x₁
↓
Calculate x₂
↓
Verify the Roots
15. Quadratic Formula vs Factorization
Both methods can be used to solve quadratic equations, but the quadratic formula is especially useful when factorization is difficult.
| Factorization | Quadratic Formula |
|---|---|
| Requires suitable factors. | Works directly using a, b and c. |
| Can be very quick for simple equations. | Useful when factorization is difficult. |
| Sign and factor selection are important. | Correct substitution and calculation are important. |
| Uses factorisation and Zero Product Property. | Uses the standard quadratic formula. |
Exam Tip: If factors are immediately visible, factorization can be quick. If factorization looks difficult, the quadratic formula provides a systematic method.
16. Quick Revision Box
Quadratic Formula
x = [−b ± √(b² − 4ac)] / 2a
Discriminant
D = b² − 4ac
Nature of Roots
D > 0 → Two distinct real roots
D = 0 → Two equal real roots
D < 0 → No real roots
Relationships Between Roots
α + β = −b/a
αβ = c/a
Exam Method
1. Standard Form
↓
2. Find a, b, c
↓
3. Find D
↓
4. Apply Formula
↓
5. Calculate Both Roots
↓
6. Verify
17. Important Questions for Board Exam
Students should practise the following question patterns:
- Solve a quadratic equation using the quadratic formula.
- Identify the values of a, b and c from a quadratic equation.
- Find the discriminant of a quadratic equation.
- Determine the nature of roots using the discriminant.
- Find equal roots of a quadratic equation.
- Find the value of a parameter when a quadratic equation has equal roots.
- Verify the relationship between roots and coefficients.
- Find the sum and product of roots.
- Solve quadratic equations having negative coefficients.
- Compare or verify answers obtained using different methods.
18. Frequently Asked Questions
Q1. What is the quadratic formula?
The quadratic formula used to find the roots of ax² + bx + c = 0 is:
x = [−b ± √(b² − 4ac)] / 2a
Q2. What is the discriminant?
The expression b² − 4ac is called the discriminant and is represented by D.
Q3. What happens when D = 0?
When D = 0, the quadratic equation has two equal real roots.
Q4. What happens when D < 0?
When D < 0, the equation has no real roots.
Q5. What is the sum of roots?
For ax² + bx + c = 0, if the roots are α and β:
α + β = −b/a
Q6. What is the product of roots?
αβ = c/a
Q7. Why is the ± sign important?
The ± sign gives two possible values, allowing us to calculate both roots of the quadratic equation.
19. Download / Study PDF
Use the PDF below for revision and practice:
Final Thoughts
The Quadratic Formula is a reliable and systematic method for solving quadratic equations. Unlike factorization, it does not require us to identify suitable factors first. Once the values of a, b and c are correctly identified, the formula can be applied directly.
The most important part is to avoid sign mistakes. Always compare the equation with ax² + bx + c = 0 and clearly write the values of a, b and c before starting the calculation.
Also remember the importance of the discriminant:
D > 0 → Two distinct real roots
D = 0 → Two equal real roots
D < 0 → No real roots
For board exams, remember the complete sequence:
Standard Form → a, b, c → D → Formula → 2 Roots → Check
Finally, revise the two important root relationships:
α + β = −b/a
αβ = c/a
With regular practice, the quadratic formula becomes a straightforward scoring method for Class 10 Maths board examinations.