Quadratic Equations – Factorization Method
Quadratic Equations are one of the most important topics in Class 10 Maths. Questions based on quadratic equations are frequently asked in board examinations and require a clear understanding of factorization, signs, splitting the middle term and the Zero Product Property.
In this chapter, we will learn how to solve quadratic equations using the Factorization Method in a simple and step-by-step way. The main goal is to convert a quadratic equation into two factors and then find the values of the variable.
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 the variable is 2
The condition a ≠ 0 is important because if a = 0, the x² term disappears and the equation becomes a linear equation.
Examples of Quadratic Equations
- x² + 5x + 6 = 0
- 2x² − 7x + 3 = 0
- 3x² − 12 = 0
- x² − 9x + 20 = 0
In all these equations, the highest power of x is 2.
2. What is the Factorization Method?
The basic idea of the factorization method is very simple. We convert the quadratic equation into the form:
(Factor 1) × (Factor 2) = 0
After factorization, we use the Zero Product Property.
Zero Product Property
If:
A × B = 0
then:
A = 0 OR B = 0
This property allows us to separate the two factors and solve them individually.
Example
Consider:
x² + 5x + 6 = 0
We need two numbers whose:
- Product = 6
- Sum = 5
The required numbers are 2 and 3.
Therefore:
x² + 2x + 3x + 6 = 0
Group the terms:
x(x + 2) + 3(x + 2) = 0
Take the common factor:
(x + 2)(x + 3) = 0
Now apply the Zero Product Property:
x + 2 = 0 or x + 3 = 0
Therefore:
x = −2, −3
3. Simple Factorization: x² + bx + c
When the coefficient of x² is 1, factorization becomes comparatively easy.
For:
x² + bx + c = 0
we need to find two numbers p and q such that:
p + q = b
and
p × q = c
Then:
x² + bx + c = (x + p)(x + q)
Example: x² − 7x + 12 = 0
Here:
- Sum = −7
- Product = 12
The required numbers are:
−3 and −4
Therefore:
x² − 7x + 12 = (x − 3)(x − 4)
So:
(x − 3)(x − 4) = 0
Using the Zero Product Property:
x − 3 = 0 → x = 3
x − 4 = 0 → x = 4
Hence:
x = 3 or 4
4. When the Coefficient of x² is Not 1
When the coefficient of x² is not 1, we generally use the splitting the middle term method.
For:
ax² + bx + c = 0
first calculate:
a × c
Then find two numbers whose:
- Product = a × c
- Sum = b
Example: 2x² + 7x + 3 = 0
Here:
a = 2, b = 7, c = 3
First calculate:
a × c = 2 × 3 = 6
Now find two numbers whose product is 6 and sum is 7.
The numbers are:
6 and 1
Split the middle term:
2x² + 6x + x + 3 = 0
Group the terms:
2x(x + 3) + 1(x + 3) = 0
Take the common factor:
(2x + 1)(x + 3) = 0
Therefore:
2x + 1 = 0
or
x + 3 = 0
Hence:
x = −1/2 or −3
5. Sign Rules – Super Important
Signs are extremely important while factorizing quadratic equations. A small sign mistake can completely change the answer.
When c is Positive
If the constant term c is positive, the two factors have the same signs.
For example:
x² + 7x + 12
We need numbers with:
- Product = +12
- Sum = +7
The numbers are:
+3 and +4
Therefore:
(x + 3)(x + 4)
Similarly:
x² − 7x + 12
requires:
−3 and −4
because:
(−3) × (−4) = +12
and:
−3 + (−4) = −7
When c is Negative
If the constant term c is negative, the factors have opposite signs.
Example:
x² + x − 6
We need two numbers whose:
- Product = −6
- Sum = +1
The numbers are:
+3 and −2
because:
3 + (−2) = 1
and:
3 × (−2) = −6
Therefore:
x² + x − 6 = (x + 3)(x − 2)
6. Step-by-Step Exam Method
In board examinations, follow a fixed process instead of trying to solve the equation randomly.
- Write the equation in standard form: ax² + bx + c = 0
- Factorize the quadratic expression.
- Apply the Zero Product Property.
- Find both values of x.
- Check the answers whenever required.
Example: x² − 5x + 6 = 0
Factorize:
(x − 2)(x − 3) = 0
Now separate the factors:
x − 2 = 0
Therefore:
x = 2
Similarly:
x − 3 = 0
Therefore:
x = 3
Hence the answer is:
x = 2, 3
7. Common Factor First
Sometimes a quadratic equation contains a common factor in all its terms. In such cases, take the common factor first.
Example: 3x² − 12x = 0
Both terms contain 3x.
Take 3x common:
3x(x − 4) = 0
Now separate the factors:
3x = 0
Therefore:
x = 0
Or:
x − 4 = 0
Therefore:
x = 4
Hence:
x = 0, 4
Exam Tip: Always check for a common factor before applying a complicated factorization method.
8. Special Cases in Factorization
Difference of Squares
One of the most useful identities is:
a² − b² = (a − b)(a + b)
Example: x² − 25 = 0
Write 25 as 5²:
x² − 5² = 0
Using the identity:
(x − 5)(x + 5) = 0
Therefore:
x − 5 = 0 → x = 5
or
x + 5 = 0 → x = −5
Hence:
x = 5, −5
Perfect Square Trinomial
The important identities are:
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Example: x² + 6x + 9 = 0
We can write:
x² + 6x + 9 = x² + 2(x)(3) + 3²
Therefore:
(x + 3)² = 0
So:
x + 3 = 0
Hence:
x = −3
9. Important Board-Level Examples
Example 1: x² + 8x + 15 = 0
We need two numbers whose sum is 8 and product is 15.
The numbers are 3 and 5.
Therefore:
(x + 3)(x + 5) = 0
So:
x = −3, −5
Example 2: x² − 9x + 20 = 0
We need two numbers whose sum is −9 and product is 20.
The numbers are −4 and −5.
Therefore:
(x − 4)(x − 5) = 0
Hence:
x = 4, 5
Example 3: 3x² − 5x − 2 = 0
Here:
a = 3, b = −5, c = −2
First calculate:
a × c = 3 × (−2) = −6
We need two numbers whose product is −6 and sum is −5.
The numbers are:
−6 and +1
Split the middle term:
3x² − 6x + x − 2 = 0
Group the terms:
3x(x − 2) + 1(x − 2) = 0
Therefore:
(3x + 1)(x − 2) = 0
Now:
3x + 1 = 0
Therefore:
x = −1/3
And:
x − 2 = 0
Therefore:
x = 2
Hence:
x = −1/3, 2
10. Common Mistakes Students Make
While solving quadratic equations through factorization, students commonly make the following mistakes:
- ❌ Forgetting to make the equation equal to zero.
- ❌ Making sign mistakes while finding factors.
- ❌ Finding factors of b instead of a × c when a ≠ 1.
- ❌ Taking only one root instead of finding both values.
- ❌ Forgetting the ± in difference of squares.
- ❌ Not checking the final answer.
- ❌ Making mistakes while splitting the middle term.
- ❌ Forgetting to apply the Zero Product Property after factorization.
Golden Rule
Factor → Separate → Solve → Check
This simple sequence can help you avoid many calculation mistakes during the examination.
11. Quick Comparison: a = 1 vs a ≠ 1
| Case | What to Find | Example |
|---|---|---|
| a = 1 | Sum = b Product = c |
x² − 5x + 6 |
| a ≠ 1 | Sum = b Product = a × c |
2x² + 7x + 3 |
Remember this distinction carefully because it is one of the most important steps in factorization-based questions.
12. Factorization Method – Complete Flow
The complete process can be remembered using the following flow:
Quadratic Equation
↓
Write in Standard Form
↓
ax² + bx + c = 0
↓
Check for Common Factor
↓
Factorize
↓
If a = 1 → Sum = b, Product = c
↓
If a ≠ 1 → Sum = b, Product = a × c
↓
Separate the Factors
↓
Each Factor = 0
↓
Find Both Roots
↓
Check the Answer
13. Important Formulas and Identities
These formulas should be remembered for quick revision:
Standard Form:
ax² + bx + c = 0
For a = 1:
Sum = b
Product = c
For a ≠ 1:
Product = a × c
Sum = b
Zero Product Property:
AB = 0 → A = 0 or B = 0
Difference of Squares:
a² − b² = (a − b)(a + b)
Perfect Square Identities:
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
14. Board Exam Quick Revision
| Concept | Remember |
|---|---|
| Quadratic Equation | ax² + bx + c = 0, a ≠ 0 |
| Factorization | Convert equation into product of factors |
| Zero Product Property | AB = 0 → A = 0 or B = 0 |
| a = 1 | Sum = b, Product = c |
| a ≠ 1 | Sum = b, Product = a × c |
| Difference of Squares | a² − b² = (a − b)(a + b) |
| Common Factor | Take it out before further factorization |
| Final Process | Factor → Separate → Solve → Check |
15. Important Questions for Board Exam
Students should practise questions based on the following patterns:
- Solve a quadratic equation by simple factorization.
- Solve a quadratic equation by splitting the middle term.
- Identify the two factors when the coefficient of x² is 1.
- Find the roots of a quadratic equation where the coefficient of x² is greater than 1.
- Solve equations involving a common factor.
- Solve quadratic equations using the difference of squares identity.
- Solve equations that form a perfect square.
- Explain the steps used in the factorization method.
16. Frequently Asked Questions
Q1. What is the standard form of a quadratic equation?
The standard form is:
ax² + bx + c = 0, where a ≠ 0.
Q2. What is the Zero Product Property?
If the product of two quantities is zero, then at least one of the quantities must be zero.
AB = 0 → A = 0 or B = 0
Q3. What should we calculate when a is not 1?
When the coefficient of x² is not 1, calculate a × c and find two numbers whose product is a × c and sum is b.
Q4. What happens when the constant term is negative?
If the constant term is negative, the two factors generally have opposite signs.
Q5. What is the easiest way to remember the factorization process?
Remember:
Factor → Separate → Solve → Check
17. Download / Study PDF
Use the PDF below for revision and practice:
Final Thoughts
Factorization is one of the most useful methods for solving quadratic equations in Class 10 Maths. The method becomes easy once you understand how to find the required numbers and how to apply the Zero Product Property.
For equations of the form x² + bx + c = 0, focus on finding two numbers whose sum is b and product is c. When the coefficient of x² is not 1, use a × c and split the middle term.
Always pay special attention to signs. Positive and negative constant terms change the signs of the factors. Also remember important identities such as the difference of squares and perfect square identities.
For board exams, keep the basic sequence in your mind:
Equation → Factorize → Each Factor = 0 → Find Roots → Check
With regular practice, factorization-based quadratic equations can become one of the quickest questions to solve in the examination.