Factorising Quadratic Expressions with Addition Factors

Objective: To learn how to factorise quadratic expressions of the form ax² + bx + c when the factors involve addition.

🟠 Warm-Up: Spot the Coefficients

A standard quadratic expression looks like:
ax² + b x + c

• a – the coefficient of x² (the number multiplying x²)
• b – the coefficient of x (the number multiplying x)
• c – the constant term (the number without any x)

Try It!

Example 1: 2x² + 7x + 3
👉 a = 2, b = 7, c = 3

Example 2: x² – 5x + 6
👉 a = 1, b = –5, c = 6 (Tip: if there is no number before x², it is 1.)

Example 3: 3x² + 8
👉 a = 3, b = 0, c = 8 (No x-term means b = 0.)

Key Point:
In every quadratic expression ax² + bx + c:
– a is with x²,
– b is with x,
– c is the number on its own.

Step 1: Recall Simple Factorisation
Before tackling full quadratics, remember how to factorise simple expressions:
x² + 5x = x(x + 5).
We take out the highest common factor (HCF).

Step 2: Factorising a Quadratic Expression
For an expression of the form x² + bx + c:
find two numbers that:
– Multiply to give c (the constant)
– Add to give b (the coefficient of x)

Write the expression as (x + m)(x + n), where m and n are the two numbers.

Example 1
x² + 7x + 12
Numbers that multiply to give us 12 and when added gives 7.

• The factors of  12 are;
• find all the factors of 12 as follows 
  

  2	12
  2	6
  3	3
  	1

•     12 + 1  = 13   not meeting the sum of 7   (2x2x3 = 12) (1)
•     6  +  2  =  8    not meeting the sum of 7   (2x3 = 6) (2)
•     4  +  3   = 7    meeting the sum of 7  (2x2 = 4)  (3), therefore these are the required factors

Therefore here 4x + 3x will be replacing 7x entirely;

  x² + 7x + 12

= x² + 4x + 3x + 12

= (x² + 4x) + (3x + 12)

= x(x + 4) + 3(x + 4)

= (x + 3)(x + 4)

___________________________

Example 2
x² + 5x + 6
Here we are looking for Numbers that multiply to give a 6 and can be added to give a 5.

the following are the factors of 6

1. 6 + 1 = 7
2. 3 + 2 = 5

as a look of things here (b) is satisfying our needs, as such 3x + 2x replaces 5x as follows;
  x² + 5x + 6

=  x² + 3x + 2x + 6

= (x² + 3x) + (2x + 6)

= x(x + 3) + 2(x + 3)

 = (x + 2)(x + 3)

Step 3: Check Your Answer by Expanding
Always multiply your factors back:
(x + 2)(x + 3) = x² + 5x + 6 ✔

Practice Questions

1. x² + 8x + 15

2. x² + 9x + 20

3. x² + 6x + 5

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Summary
• A quadratic expression can be written in the form ax² + bx + c.
• To factorise when a = 1, look for two numbers whose product is c and whose sum is b.
• Always expand your factors to check.

Next Lesson: Factorising quadratics where a ≠ 1 (e.g. 2x² + 5x + 3).