1. Factoring
ax2 + bx + c
“Bottoms Up”

2. 6x2 + 13x + 6
Step 1: multiply the constant (c) term by the coefficient (a), of the leading term
Constant is 6
Leading term is 6
Therefore 6 * 6 is 36

3. 6x2 + 13x + 6
Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.
x2 + 13x + 36

4. Step 3: Factor the new equation from
step 2: x2 + 13x +36
(x + 4)(x + 9)

5. (x + 4)(x + 9)
Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.
Original coefficient of the leading term: 6

6. Then reduce the fractions:

7. Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:

8. Check it- use FOIL:

9. Remember you may check some equations using graphing calculator- type in the equation in y = and find where it crosses the x-axis. Not all equations cross the x-axis. This particular one crosses at x = -2/3 and x = -3/2. Look at these solutions and the end result of step 4, and then look at step 5. Factoring leads to solutions, the zeros, the roots.

10. 10x2 - x - 3
Step 1: multiply the constant (c) term by the coefficient (a), of the leading term
Constant is -3
Leading term is 10
Therefore -3 * 10 is -30

11. 10x2 - x - 3
Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.
x2 - x - 30

12. (x + 5)(x - 6) x2 - x - 30 Step 3: Factor the new equation from
step 2: x2 - x - 30
(x + 5)(x - 6)

13. (x + 5)(x - 6)
Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.
Original coefficient of the leading term: 10
(x + 5)(x - 6)

14. Then reduce the fractions:
(x + 1)(x - 3)

15. Step 5: Now we use the “bottoms up” method- Move the denominator into the numerator to get:
(2x + 1)(5x – 3)
(x + 1)(x - 3)

16. (2x + 1)(5x – 3)
Check it- use FOIL:
10x2 -6x + 5x – 3
10x2 - x – 3

17. You Try!
6x2 - 2x – 28 Step 1: multiply the constant (c) term by the coefficient (a), of the leading term Constant is -28 Leading term is 6 Therefore 6 * -28 is -168

18. 6x2 - 2x – 28
Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.
x2 -2x -168

19. x2 -2x -168 Step 3: Factor the new equation ( x -14 )(x +12 )
Factors of -168
Sum of factors
-2, -84 no
-4, 42
-8, 21 13
-12, 14 2
Switch
-14, 12 -2

20. ( x -14 )(x +12 )
Step 4: Divide each constant term in the factored form by the original coefficient of the leading term.
Original coefficient of the leading term: 6
( x -14 )(x +12 )

21. ( x -14 )(x +12 )
Then reduce the fractions:
( x -7 )(x +2 )
Step 5: Now we use the “bottoms up”

22. ( 3x -7 )(x +2 ) Check your answer with factoring (your choice).
Conversation with a caution
6x2 - 2x – 28
Purpose of factoring

23. Quick steps! Step 1: multiply (c) (a),
Step 2: Rewrite the equation by replacing the constant with the number in step 1, and remove the leading coefficient.
Step 3: Factor the new equation
Step 4: Divide each constant term in the factored form by the original coefficient of the leading term and then reduce the fractions:
Step 5: Now we use the “bottoms up”