# Factorizing quadratic expressions

**Quadratic expressions**

**Quadratic expressions **are expressions where by the highest power of the variable is 2.

For example,

*x*² – 2

*w² + 3w + 1 *

4 – 5*g²*

The general form of a quadratic expression in *x* is:

*ax² + bx² + c (a ≠ 0)*

*x* = **variable**.

*a* = constant and is the **coefficient** of *x*²

*b* = constant and is the **coefficient** of *x*.

*c* = constant

**Factorizing expressions**

*(a + 1)(a + 2)* → *a² + 3a + 2 *(expanding or multiplying out)

*a² + 3a + 2 → (a + 1)(a + 2) *(factorising)

Points to note:

- When an expression is expanded – the brackets are removed
- When an expression if factorised – the brackets are written within it

**Factorizing quadratic expressions**

*x²* + *bx* + *c* is factorised by being written as (*x* + *d*)(*x* + *e*)

*d* and *e =* integers

(*x* + *d*)(*x* + *e*) is expanded by:

=(*x* + *d*)(*x* + *e*) = *x²* + *dx + e**x* + *d**e*

= *x²* + (*d + e)**x* + *d**e*

Points to note:

*d +**e = b (coefficient of x)*- The product of
*d*and*e*=*c*(constant)

**Factorising quadratic expressions 2**

*ax²* + *bx* + *c* is factorised by being written as (d*x* + *e*)(f*x* + *g*)

*d**,** e**,** f* & *g* = integers.

(*dx* + *e*)(*fx* + *g*) is expanded by:

(*dx* + *e*)(*fx* + *g*) = *dfx²* + dg*x + **e**fx* + *e**g*

= *dfx²* + (dg* + **e**f)x* + *e**g*

Points to note:

*a = df**b = (dg +ef)**c = eg*

**Factorizing the difference between two squares**

The quadratic expression * x² – a² *is called “the difference between two squares”

This is factorised as follows:

*x²* + a*²* = (*x* + *a*)(x – *a*)

Therefore:

*9x²* + 16 = (3*x* + *4*)(3x – *4*)