Factoring Quadratic Trinomials with Leading Coefficient Other Than One
(1)
Find the product (a x c)
(2)
Write down the sum (b)
(3)
List all of the possible factors of the result from step 1.
(4)
Determine which factors will add together to give the sum from step 2.
(5)
Rewrite the expression replacing the middle term with the sum of factors from step 4.
(6)
Complete factorisation by grouping.
Example (1)
2x2 + 8x + 8
(1)
Product = 2 x 8 = 16
(2)
Sum = 8
(3)
Factors of 16: ( 1, 16), (-1, - 16), (4, 4), (-4, -4), (2, 8), (-2, -8)
(4)
(4, 4) (this combination was chosen because 4 + 4 = 8)
(5)
2x2 + 4x + 4x + 8
(6)
(2x2 + 4x) + (4x + 8) = 2x(x + 2) + 4(x + 2) = (2x + 4)(x + 2)
Example (2)
2x2 + 5x – 12
1)
Product = 2 x -12 = -24
(2)
Sum = 5
(3)
Factors of 16: ( 1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), (-4, 6).
(4)
(-3, 8) (this combination was chosen because -3 + 8 = 5)
(5)
2x2 – 3x+ 8x – 12
(6)
(2x2 – 3x) + (8x – 12) =
Factorise the following:
(1)
2x2 + 5x + 2
(2)
3x2 – 10x – 8
(3)
4x2 – 4x + 1
(4)
5x2 + 4x – 1
1.
Factorise completely
a 2  ab  ac  bc
2.
Factorise completely
x2  y 2  4 x  4 y
3.
Factorise
3a  at  6 p  2 pt
4. Factorise completely
i. 1  9x2
ii. 3x2  7 x  6
5. Factorise completely 3x 2  21x
i. 4a2  1
ii. 6 x2  x  2
6. Factorise completely
i. 9a 2  b2
ii. 3x  8 y  4 xy  6
1.
7. Factorise completely
i. 9  25m2
ii. 2 x2  x  15
iii. x  y  ax  ay
8. Factorise completely
12 p3q  8 p 2 q3
9. Factorise completely
i. 6  a 2
ii. 5x  xy  2 y  10
10. Factorise completely
i. x2  xy
ii. e2  1
iii. 5 p 2  9 pq  2q 2
11. Factorise completely
i. 4g 2  f 2
ii. tm  3t  2 pm  6 p
12. Factorise completely
i. 8h2  4h
ii. 4a2  1