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Factorizing Quadratic
Equations
06/07/2017
1
x2 ±
bx ± c
If this is positive, both signs are the same.
If this is positive,
both signs are +.
The numbers ADD to give this
value and MULTIPLY to give
this value.
If this is negative,
both signs are -.
If this is negative, both signs are
different.
The numbers have a
DIFFERENCE of this value
The largest value takes this sign.
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2
eg.
x2 +
5x + 4
This is positive, so both signs are the same.
This is positive, so
both signs are +.
Answer
06/07/2017
The numbers ADD to give 5
and MULTIPLY to give 4.
In other words 4 and 1.
(x+4)(x+1)
3
eg.
x2 -
10x + 16
This is positive, so both signs are the same.
The numbers ADD to give 10 and MULTIPLY
to give 16.In other words 2 and 8.
This is negative, so
both signs are -.
Answer
06/07/2017
(x-2)(x-8)
4
x2 -
eg.
6x - 16
This is negative, so both signs are
different.
The numbers MULTIPLY to
give this value and have a
DIFFERENCE of this value. In
other words 2 and 8.
The largest value takes this sign.
Answer
06/07/2017
(x+2)(x-8)
5
eg.
x2 +
4x
- 32
This is negative, so both signs are
different.
The numbers MULTIPLY to
give this value and have a
DIFFERENCE of this value. In
other words 4 and 8.
The largest value takes this sign.
Answer
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(x-4)(x+8)
6
1. x2 + 6x + 5
Exercise
1. (x+5)(x+1)
2. x2 - 6x + 5
2. (x-5)(x-1)
3. x2 + 8x + 16
3. (x+4)(x+4)
4. x2 - 10x + 16
4. (x-8)(x-2)
5. x2 - 4x – 12
5. (x-6)(x+2)
6. x2 + 6x – 16
6. (x+8)(x-2)
7. x2 + 15x – 16
7. (x+16)(x-1)
8. x2 – 7x -30
8. (x-10)(x+3)
9. x2 + x – 20
9. (x+5)(x-4)
10. x2 – 16
10.(x+4)(x-4)
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eg.
x2 ±
0x
- 16
This is negative, so both signs are
different.
The numbers MULTIPLY to
give this value and have a
DIFFERENCE of this value. In
other words 4 and 4.
The largest value takes this sign.
Irrelevant in this case!
Answer
06/07/2017
(x+4)(x-4)
8
What happens when there is more
than one lot of x2, i.e. the general
case of ax2 ± bx ± c
There is a slight change here.
First of all multiply a and c.
We are now looking for 2 values that multiply to give (a x c)
and either add to give, or have a difference of b.
We must now rewrite the equation and look to factorise the
two separate parts of the equation to give a common factor.
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9
eg.
2
2x
+
9x + 4
(2x4 =+8)
This is positive, so both signs are the same.
This is positive, so
both signs are +.
Rewrite
Factorize
Answer
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The numbers ADD to give 9
and MULTIPLY to give 8.
In other words 8 and 1.
2x2 +8x +1x + 4
2x(x + 4) +1(x + 4)
(2x+1)(x + 4)
10
eg.
2
3x
-
5x - 2
(3x-2=-6)
This is negative, so the signs are different.
This is negative, so
the largest value is -.
Rewrite
Factorize
Answer
06/07/2017
The numbers HAVE A
DIFFERENCE OF 5 and
MULTIPLY to give 6.
In other words 6 and 1.
3x2 -6x +1x - 2
3x(x - 2) +1(x - 2)
(3x+1)(x-2)
11
eg.
2
8x
-
10x - 3
(8 x -3=-24)
This is negative, so the signs are different.
This is negative, so
the largest value is -.
Rewrite
Factorize
Answer
06/07/2017
The numbers HAVE A
DIFFERENCE OF 10 and
MULTIPLY to give 24.
In other words 12 and 2.
8x2 -12x +2x - 3
4x(2x - 3) +1(2x - 3)
(4x+1)(2x-3)
12
1. 2x2 + 5x + 3
Exercise
1. (2x+3)(x+1)
2. 2x2 + 7x + 3
2. (2x+1)(x+3)
3. 3x2 + 7x + 2
3. (3x+1)(x+2)
4. 2x2 - x - 15
4. (2x+5)(x-3)
5. 2x2 + x – 21
5. (2x+7)(x-3)
6. 3x2 - 17x – 28
6. (3x+4)(x-7)
7. 6x2 + 7x - 3
7. (2x+3)(3x-1)
8. 10x2 + 9x + 2
8. (5x+2)(2x+1)
9. 12x2 + 23x + 10
9. (3x+2)(4x+5)
10. 6x2 – 27x + 30
10.3(2x-5)(x-2)
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