Download How to factorise quadratics

How to factorise quadratics
Reminder: how to expand brackets
How to factorise quadratics
4x(2 + x)
= 8x + 4x2
A quadratic expression is one that contains an x2 term
(and no other higher powers of x).
Single brackets
Multiply each term inside the
brackets by the term outside
(x – 5)(x + 2)
= x2 + 2x – 5x – 10
= x2 – 3x – 10
You will need to know how to expand and factorise
quadratic expressions.
Double brackets
Multiply each term in one set by
each term in the other set
(x – 3)2
= (x – 3)(x – 3)
= x2 – 3x – 3x + 9
= x2 – 6x + 9
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Reminder: how to expand brackets
How to factorise quadratics
Try these:
Step one – common factors
1. x(3x – 5)
= 3x2 – 5x
x2 – 3x
= x( x – 3 )
3. (x – 4)(x + 5) = x2 + x – 20
= 6x2 – 18x
5. (2x – 1)(x – 2) = 2x2 – 5x + 2
6. (x – 4)2
4x2 – 2x
= 2x( 2x – 1 )
= x2 – 8x + 16
Factorising is the opposite of expanding – where an
expression is put into brackets.
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x is a factor of x2 and 3 x. Take this
term outside the brackets.
Find the factor pair for each term and
put it inside the bracket. Watch your
signs!
Remember to check for the highest
common factor. Both terms have 2 and
x as a factor, so 2x comes outside the
brackets.
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How to factorise quadratics
How to factorise quadratics
Step one – common factors
Step two – double brackets
Try these:
1. x2 + 5x
= x(x + 5)
2. 3x2 + 6x
= 3x(x + 2)
3. 2x2 – 4x + 8
= 2(x2 – 2x + 4)
4. x2 + 4x
= x(x + 4)
5. 4x2 – 12y
= 4(x2 – 3y)
6. 3x2 – 6x – 9
= 3(x2 – 2x – 3)
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Expanding two brackets often gives a trinomial
(expression with three terms):
(x + 5)(x + 2) = x2 + 5x + 2x + 10
= x2 + 7x + 10
How was the constant (10) calculated?
The two number terms in each bracket were
multiplied together.
How was the x-coefficient (7) calculated?
The two number terms in each bracket were
added together.
7. 5x2(a – b) – 2y2(a – b) = (a – b)(5x2 – 2y2)
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If every term in the expression has a common factor,
you can use single brackets:
2. (x – 3)(x + 4) = x2 + x – 12
4. 3x(2x – 6)
Squared brackets
Multiply by itself, like double
brackets
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How to factorise quadratics
How to factorise quadratics
How to factorise quadratics
Step two – double brackets
Step two – double brackets
So to factorise a trinomial, we need to find two numbers
which ...
So to factorise a trinomial, we need to find two numbers
which ...
• multiply to give the constant
• add to give the x-coefficient.
• multiply to give the constant
• add to give the x-coefficient.
x2 + 7x + 6
2×3
2+3=5
1×6
1+6=7
Try these:
= (x + 1)(x + 6)
Check your answers by expanding the brackets again:
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= (x – 1)(x + 4)
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Step two – double brackets
1×6
-1 × -6
2×3
-2 × -3
1+6=7
-1 + -6 = -7
2+3=5
-2 + -3 = -5
2 × -2
1 × -4
-1 × 4
2 + -2 = 0
1 + -4 = -3
-1 + 4 = 3
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1. x2 – 7x + 12
= (x – 3)(x – 4)
2. x2 – 5x – 6
= (x + 1)(x – 6)
3. x2 + 5x – 6
= (x – 1)(x + 6)
4. x2 – 2x – 3
= (x – 3)(x + 1)
5. x2 + 9x + 14
= (x + 7)(x + 2)
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How to factorise quadratics
How to factorise quadratics
Special case
Special case – difference of two squares
Try these:
1. (x + 3)(x – 3)
= x2 – 9
2. (x – 4)(x + 4)
=
3. (2x + y)(2 x – y) =
x2
To use this method, the expression must take this form:
square
– 16
4x2
–
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another square
And your solution will take this form:
(x + 3)(x – 3)
two brackets
All these answers show a difference of two
squares.
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subtract
x2 – 9
y2
What is interesting about these three expressions?
There is no x term. Can you explain why?
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Check for
common
factors
first!
Try these:
If the trinomial contains a negative sign, you’ll need to take
that into account too.
x2 – 3x – 4
= (x + 4)(x + 2)
How to factorise quadratics
Step two – double brackets
= (x – 2)(x – 3)
2. x2 + 6x + 8
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How to factorise quadratics
x2 – 5x + 6
= (x + 3)(x + 2)
3. 2x2 + 12x + 10 = 2(x2 + 6x + 5)
= 2(x + 5)(x + 1)
(x + 1)(x + 6)= x2 + x + 6x + 7
= x2 + 7x + 6
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1. x2 + 5x + 6
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one add,
one
subtract
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square root
each term
12
2
How to factorise quadratics
How to factorise quadratics
How to factorise quadratics
Special case – difference of two squares
Try these:
Mixture
Try these:
1. x2 – 25
= (x + 5)(x – 5)
2. 81 – y2
= (9 + y)(9 – y)
1. x2 + 4x – 21
= (x + 7)(x – 3)
3. 4x2 – 9
= (2x + 3)(2x – 3)
2. x2 – 4x – 12
= (x + 2)(x – 6)
+9
=
3(x2
+ 3)
4. 9x2 – 16y2
= (3x + 4y)(3x – 4y)
3.
5. 3 – 12x2
= 3(1 – 4 x2)
4. 2x2 + 18x + 40 = 2(x + 4)(x + 5)
= 3(1 + 2x)(1 – 2x)
5. 4x2y2 – 36x2
What about x2 + 4?
3x2
Check for
common
factors
first!
= 4x2(y + 2)(y – 2)
We can’t do it as it’s not a difference!
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How to factorise quadratics
Summary
Use your solutions to complete these sentences:
With any expression, you should first check if ...
... there are any common factors.
When factorising into two brackets, if the constant term is
positive then ...
... its factors will both be positive or both be
negative.
When factorising into two brackets, if the constant term is
negative then ...
... its factors will be one positive and one
negative.
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