Factorizing Quadratic Equations

1
Factorizing Quadratic Equations
2
x2 bx c
If this is positive, both signs are the same.
The numbers ADD to give this value and MULTIPLY
to give this value.
If this is positive, both signs are .
If this is negative, both signs are different.
If this is negative, both signs are -.
The numbers have a DIFFERENCE of this value
The largest value takes this sign.
3
eg. x2 5x 4
This is positive, so both signs are the same.
The numbers ADD to give 5 and MULTIPLY to give
4. In other words 4 and 1.
This is positive, so both signs are .
Answer (x4)(x1)
4
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 (x-2)(x-8)
5
eg. x2 - 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 (x2)(x-8)
6
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 (x-4)(x8)
7
Exercise

• x2 6x 5
• x2 - 6x 5
• x2 8x 16
• x2 - 10x 16
• x2 - 4x 12
• x2 6x 16
• x2 15x 16
• x2 7x -30
• x2 x 20
• x2 16

• (x5)(x1)
• (x-5)(x-1)
• (x4)(x4)
• (x-8)(x-2)
• (x-6)(x2)
• (x8)(x-2)
• (x16)(x-1)
• (x-10)(x3)
• (x5)(x-4)
• (x4)(x-4)

8
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 (x4)(x-4)
9
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.

10
eg. 2x2 9x 4 (2x4 8)
This is positive, so both signs are the same.
The numbers ADD to give 9 and MULTIPLY to give
8. In other words 8 and 1.
This is positive, so both signs are .
Rewrite 2x2 8x 1x 4
Factorize 2x(x 4) 1(x 4)
Answer (2x1)(x 4)
11
eg. 3x2 - 5x - 2 (3x-2-6)
This is negative, so the signs are different.
The numbers HAVE A DIFFERENCE OF 5 and MULTIPLY
to give 6. In other words 6 and 1.
This is negative, so the largest value is -.
Rewrite 3x2 -6x 1x - 2
Factorize 3x(x - 2) 1(x - 2)
Answer (3x1)(x-2)
12
eg. 8x2 - 10x - 3 (8 x -3-24)
This is negative, so the signs are different.
The numbers HAVE A DIFFERENCE OF 10 and MULTIPLY
to give 24. In other words 12 and 2.
This is negative, so the largest value is -.
Rewrite 8x2 -12x 2x - 3
Factorize 4x(2x - 3) 1(2x - 3)
Answer (4x1)(2x-3)
13
Exercise

• 2x2 5x 3
• 2x2 7x 3
• 3x2 7x 2
• 2x2 - x - 15
• 2x2 x 21
• 3x2 - 17x 28
• 6x2 7x - 3
• 10x2 9x 2
• 12x2 23x 10
• 6x2 27x 30

• (2x3)(x1)
• (2x1)(x3)
• (3x1)(x2)
• (2x5)(x-3)
• (2x7)(x-3)
• (3x4)(x-7)
• (2x3)(3x-1)
• (5x2)(2x1)
• (3x2)(4x5)
• (2x-5)(3x-6)