Definitions

• Factorising: the opposite of expanding; it means "putting into brackets".
• Factors: parts of a number that divide exactly into it. For example, 5 divides into 10.
• Quadratic - an expression which has a degree of 2
• Degree- the highest power of a variable in an expression

Enter the world of Quadratics!

Quadratics have the form \(ax^2+bx+c\), where a, b and c are integers.

It's just a special type of expression, which has an x^2 term as the highest power.

Like normal expressions we can expand them and factorise them.

Recall that:

Odd number of minus signs → negative result

Even number of minus signs → positive result

Factorising Quadratics Where a=1

For simple quadratics where a =1, it is a simple procedure. Just remember LASEr

• LIST: Find all the factors of c, listing them starting from 1 to c. Stop listing when factors start to repeat. See the example to understand.
• ADD/SUBTRACT: Beside each pair of factors, check whether these factors add or subtract to give b.
• SET UP: Once you've found a pair that factors c and sums to b, write your brackets \((x + a)(x + b)\).
• EXPAND: Check if it expands back to the original quadratic expression.

Example 1

\(x^2+5x+6\)

LIST Factors of c

• 1,6
• 2,3
• 3,2 <--STOP HERE and ignore, as the factors repeat.

ADD/SUBTRACT -Find the sum of the factos to make b

b is 5. So for each pair of factors try to make 5.

• 1+6=7, so that's not it
• 6-1=5 that's not correct, because, remmeber, the sign of c is a positve and recall that + x - = -. So that would mean c would be -6 not +6, where not looking for =6 though, so ignore.
• 2+3 = 5 this is correct because they are both postive and sum to 5 and multiply to give 6.

SET UP

So the pair you need is (+3,+2) now place them in brackets. You get \((x+3)(x+2)\)

This is the solution. You can expand to check but I'll leave that as homeweork for you once you cover expanding double brackets.