Factoring Trinomial of the Form ax2 + bx + c for a ≠1
"The Hard Case"
Recap: "The Easy Case" a = 1
Example 1 Factor the following
x2 x 30
x2 + 7x – 6
Note: example 2 is "prime" or unfactorable over the integers
Rules for Sign for factoring trinomials:
a) If c is positive (+ve), both factors have the same sign. The sign is
the same as the coefficient b.
a) x2 + 5x + 6 b) x2 5x + 6
b) If c is negative (ve), both factors have opposite signs where the
larger factor has the same sign as the coefficient b.
c) x2 + 4x 12 d) x2 4x 12
Example 5 Factor 2x2 + x – 6
1) Remove any common factors from the equation (ie numbers or
variables) (1 is the only common factors in this example.)
2) Multiply the factors "a" and "c" . Find the factors of the product "ac" that
add to b
Here a = 2 and c = 6 so ac = 12
12 has factors. 1, 12 We must choose the factors that
2, 6 add to + 1
3) Draw a box and put the first term in the upper lefthand
box. Put the last term in the lower righthand box.
4) Now take the factors and put them, complete
with their signs and variables, in the
diagonal corners. It does not matter which box
you put the factors in you will get the same answer.)
(Our factors are 3, 4. The variable is x.)
+2x2 + 4 x
3 x 6
5) Find the common factors from each row and each column. Use the
sign from the nearest box as the sign for your factor.
+2x2 + 4 x
3 x 6
6) Read the factors from across the top and down the left of the box.
Therefore our factors are 2x2 + x – 6 = (2x – 3)(x + 2).
Example 6 Factor the following:
a) 2x2 + 7x +3 b) 12x2 + x 6
c) 3x2 5x 12 d) 2x2 + 11x 5
e) 2x2 6 x +4