Double brackets consist of two binomial expressions multiplied together, e.g. (x+1)(y−3). You should be able to multiply out a set of double brackets as well as factorise back into double brackets.
Multiplying out double brackets
When multiplying out double brackets, make sure to multiply every term in the first bracket with every term in the second bracket, you can use F.O.I.L., or a multiplication grid.
FOIL
First
Multiply the first terms of each bracket.
Outside
Multiply the outside terms.
Inside
Multiply the two inside terms.
Last
Multiply the last terms of each bracket.
Note:Once the brackets have been multiplied out, simplify the terms where possible.
Factorising a quadratic means putting it into double brackets. A quadratic is in the form
ax2+bx+c
When a=1 we can factorise a quadratic as follows.
Procedure
1.
Ensure the quadratic is in the form ax2+bx+c and identify a,b and c
2.
Set up the answer with two brackets and an x at the start of each bracket, as follows (x)(x)
3.
Find two numbers which multiply to give c and add to give b
4.
Fill these numbers in the double brackets
Example 3
Factorise
x2+10x+24
The quadratic is in the correct form whereb=10andc=24.
6×4=24and6+4=10, so our numbers are6and4.
x2+10x+24=(x+6)(x+4)
Solving a quadratic
If the quadratic is in an equation, rearrange it so it is equal to 0, then it can be solved for x.
Example 4
Solve
x2+4x−21=0
First factorise the quadratic, then the solution is the number in the bracket with the sign reversed. This is because this value ofxwill make the bracket equal to0.
x2+4x−21(x+7)(x−3)x=−7,x=3=0=0
Difference of two squares
The difference of two squares is a where you have one term squared minus another term squared. The expression can be factorised into double brackets.
x2−y2=(x+y)(x−y)
Example 5
Factorise9a2−4b2
9a2is the square of3a. 4b2is the square of2b. Write3aas the first term in both brackets and2bas the last term in both brackets. One bracket should have addition, the other should have subtraction.
9a2−4b2=(3a+2b)(3a−2b)
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Unit 1
Single brackets: Expanding and factorising
Unit 2
Expanding double brackets: FOIL and multiplication grid
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Unit 3
Double brackets: Expanding and factorising
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FAQs - Frequently Asked Questions
How do you factorise into double brackets?
Put the quadratic into the form ax^2 + bx + c. If a=1, then factorise the quadratic by finding two numbers that multiply to give c, and add to give b. These two numbers will go into the double brackets. e.g. x^2 + 7x + 10 = (x+2)(x+5).
How do you multiply out brackets?
You can use FOIL (First, outside, inside, last) or a multiplication grid to multiply out double brackets. Remember that each term in the first bracket should be multiplied by each term in the second bracket.
What do double brackets mean?
Double brackets consist of two binomial expressions multiplied together, e.g. (x+1)(y-3). Each term in the first bracket should be multiplied by each term in the second bracket.