Factorising Quadratic Expressions When a ≠ 1

Objective:
To learn how to factorise quadratic expressions of the form ax² + bx + c when a is greater than 1.

🟢 Step 1: Recall the Standard Form

A quadratic expression looks like:
ax² + bx + c

In the last lesson, we learned how to factorise when a = 1.
Now we will see what to do when a ≠ 1, for example 2x² + 5x + 3.

🔹 Step 2: Multiply a × c

We start by multiplying the first and last numbers (a × c).

Example:
2x² + 5x + 3
→ a = 2, b = 5, c = 3
→ a × c = 2 × 3 = 6

Now find two numbers that:

• multiply to give a × c = 6

• add to give b = 5

• using the previous factoring methods from previous lesson 3 we can safely get the following factors
• 6 + 1 = 7
• 3 + 2 = 5

✅ The numbers are 2 and 3 (why? because they add up to 5 )

🔹 Step 3: Split the Middle Term

Replace the middle term (5x) with two terms using these numbers.

2x² + 2x + 3x + 3

Now we have four terms instead of three.

🔹 Step 4: Group the Terms

Group the first two and the last two terms:

(2x² + 2x) + (3x + 3)

Factorise each group separately:

2x(x + 1) + 3(x + 1)

🔹 Step 5: Factorise the Common Bracket

Both groups contain (x + 1), so we can factor that out:

(2x + 3)(x + 1)

✅ Therefore,
2x² + 5x + 3 = (2x + 3)(x + 1)

🟣 Example 2

3x² + 10x + 8

a × c = 3 × 8 = 24

factors of 24 

24 + 1 = 25

12 + 2 = 14

8 + 3 = 11

6 + 4 = 10

Numbers that multiply to 24 and add to 10 → 4 and 6

Replace the middle term (1ox with the the two correct factors 4x and 6x):
3x² + 4x + 6x + 8

Group: (3x² + 4x) + (6x + 8)

Factorise: x(3x + 4) + 2(3x + 4)

Final factors: (x + 2)(3x + 4)

✅ Check: (x + 2)(3x + 4) = 3x² + 10x + 8 ✔

🟢 Step 6: Practice Questions

Try to factorise the following:

1. 2x² + 7x + 3

2. 3x² + 8x + 4

3. 4x² + 11x + 6

4. 5x² + 9x + 4

5. 156y² + 25y + 1

Hint: Multiply a × c, find two numbers that multiply to a×c and add to b, then split the middle term.

🟠 Summary

• When a ≠ 1, multiply a × c first.

• Find two numbers whose product is a×c and whose sum is b.

• Split the middle term and group terms in pairs.

• Factorise each group and take out the common bracket.

• Expand to check your final answer.

Next Lesson:
Factorising Quadratic Expressions with Negative or Mixed Signs (e.g. 2x² – 5x – 3)