Quadratic expressions follow the pattern, Ax^{2} + Bx + C.

We shall be looking at simple quadratic expressions, namely where A = 1, which will follow the pattern, x^{2} + Bx + C. When this expression is factorised, the result will be the product of two brackets.

x^{2} + Bx + C = ( x ) ( x )

**Signs**First we need to work out what the signs in the brackets will be. We look at the constant C, if it is POSITIVE, then both the signs in the brackets will be the SAME. We then look at B, if it is POSITIVE then both the signs in the brackets will be POSITIVE, if it is NEGATIVE both the signs in the brackets will be NEGATIVE.

x^{2} + Bx +** **C = ( x + ) ( x + )

x^{2} - Bx +** **C = ( x - ) ( x - )

But if C is NEGATIVE then the signs in the brackets will be DIFFERENT.

x^{2} + Bx -** **C = ( x + ) ( x - ) or ( x - ) ( x + )

x^{2} - Bx -** **C = ( x + ) ( x - ) or ( x - ) ( x + )

**Numbers**Now to work out the numbers that go into the brackets.

**If C is POSITIVE**then the missing numbers are the two FACTORS of C that ADD together to make B.

e.g. x^{2} + 5x +** **6

C is positive, so both the signs will be the same. B is positive so both the signs will be positive

So x^{2} + 5x +** **6 = ( x + ) ( x + )

The two factors of 6 that add up to 5 are 3 and 2

So x^{2} + 5x +** **6 = ( x + 3 ) ( x + 2 )

e.g. x^{2} - 5x +** **4

C is positive, so both the signs will be the same. B is negative so both the signs will be negative

So x^{2} - 5x +** **4 = ( x - ) ( x - )

The two factors of 4 that add up to 5 are 4 and 1

So x^{2} - 5x +** **4 = ( x - 4 ) ( x - 1 )

Have a go at these:

x^{2} + 7x +** **6 x^{2} - 10x +** **21 x^{2} - 13x +** **12 x^{2} + 9x +** **20

**If C is NEGATIVE** then the missing numbers are the two FACTORS of C that have a DIFFERENCE of B. **ALSO** if B is POSITIVE then the larger number goes with the POSITIVE sign, if B is NEGATIVE then the larger number goes with the NEGATIVE sign

e.g. x^{2} + 4x -** **12

C is negative, so both the signs will be different.

So x^{2} + 4x -** **12 = ( x + ) ( x - )

The two factors of 12 that have a difference of 4 are 6 and 2. B is positive so the 6 goes with the positive sign

So x^{2} + 4x -** **12 = ( x + 6 ) (x - 2 )

Compared with x^{2} - 4x -** **12 = ( x + 2 ) ( x - 6 )

Have a go at these:

x^{2} + x -** **6 x^{2} - 5x -** **24 x^{2} + 8x -** **9 x^{2} - 5x -** **14

**Answers:**x

^{2}+ 7x +

**6 = ( x + 1 ) ( x + 6 )**

x

^{2}- 10x +

**21 = ( x - 3 ) ( x - 7 )**

x

^{2}- 13x +

**12 = ( x - 1 ) ( x - 12 )**

x

^{2}+ 9x +

**20 = ( x + 4 ) ( x + 5 )**

x

^{2}+ x -

**6 = ( x + 6 ) ( x - 1 )**

x

^{2}- 5x -

**24 = ( x - 8 ) ( x + 3 )**

x

^{2}+8 x -

**9 = ( x + 9 ) ( x - 1 )**

x

^{2}+ 5x -

**14 = ( x - 7 ) ( x + 2 )**