The Diamond Method of Factoring a Quadrac Equaon
Important:
Remember that the rst step in any factoring is to look at each term and factor out the greatest
common factor. For example: 3x
2
+ 6x + 12 = 3(x
2
+ 2x + 4) AND 5x
2
+ 10x = 5x(x + 2)
If the leading coecient is negave, always factor out the negave. For example:
-2x
2
- x + 1 = -1(2x
2
+ x - 1) = -(2x
2
+ x - 1)
Using the Diamond Method:
Example 1
Factor 2x
2
+ 11x + 15 using the Diamond Method.
Step 1: Mulply the coecient of the x
2
term (+2) and the constant (+15) and
place this product (+30) in the top quarter of a large “X.”
Step 2: Place the coecient of the middle term in the boom quarter of the
“X.” (+11)
Step 3: List all factors of the number in the top quarter of the “X.”
+30
(+1)(+30) (-1)(-30)
(+2)(+15) (-2)(-15)
(+3)(+10) (-3)(-10)
(+5)(+6) (-5)(-6)
Step 4: Idenfy the two factors whose sum gives the number in the boom
quarter of the “x.” (5 ∙ 6 = 30 and 5 + 6 = 11) and place these factors in
the le and right quarters of the “X” (order is not important).
Step 5: Break the middle term of the original trinomial into the sum of two terms
formed using the right and le quarters of the “X.” That is, write the rst
term of the original equaon, 2x
2
, then write 11x as + 5x + 6x (the num
bers from the “X”), and nally write the last term of the original equaon,
+15 , to get the following 4-term polynomial:
2x
2
+ 11x + 15 = 2x
2
+ 5x + 6x + 15
+30
+11
+6
+5
+30
+11
Step 6: Factor by Grouping:
Group the rst two terms together and the last two terms together. 2x
2
+ 5x + 6x + 15
Factor out common factors from each group. x(2x + 5) + 3(2x + 5)
Factor out the common binomial factor. (2x + 5)(x + 3)
Example 2
Factor 12x
2
- 5x - 2 using the Diamond Method.
Step 1: Mulply the coecient of the x
2
term (+12) and the constant (-2) and
place this product (-24) in the top quarter of a large “X.”
Step 2: Place the coecient of the middle term in the boom of the “X.” (-5)
Step 3: List all factors of -24: -24
(+1)(-24) (-1)(+24)
(+2)(-12) (-2)(+12)
(+3)(-8) (-3)(+8)
(+4)(-6) (-4)(+6)
Step 4: Idenfy the factors whose sum is –5: (+3 ∙ -8 = -24 and 3 - 8 = -5) and
place them in the le and right quarters of the “X” (order is not important).
-24
-5
-8
+3
-24
-5
Step 5: Break the middle term of the original trinomial into the sum of two terms formed using the right and
le quarters of the “X.” 12x
2
- 5x - 2 = 12x
2
+ 3x - 8x - 2
Step 6: Factor by Grouping:
Group the rst two terms together and the last two terms together. 12x
2
+ 3x - 8x - 2
Factor out common factors from each group. (Factor out negave.) 3x(4x + 1) - 2(4x + 1)
Factor out the common binomial factor. (4x + 1)(3x - 2)
Example 3
Factor 16x
2
- 26x + 3 using the Diamond Method.
Step 1: (+48) Step 2: (-26)
Step 3: List all factors of 48: +48
(+1)(+48) (-1)(-48)
(+2)(+24) (-2)(-24)
(+3)(+16) (-3)(-16)
(+4)(+12) (-4)(-12)
(+6)(+8) (-6)(-8)
Step 4: Idenfy the factors whose sum is -26: - 2 - 24 = - 26
Step 5: 16x
2
- 26x + 3 = 16x
2
- 2x - 24x + 3
Step 6: Factor by Grouping:
Group the rst two terms together and the last two terms together. 16x
2
- 2x - 24x + 3
Factor out common factors from each group. (Factor out negave.) 2x(8x - 1) - 3(8x - 1)
Factor out the common binomial factor. (8x - 1)(2x - 3)
-24
-2
+48
-26