This document provides instructions and examples for solving "riddles" involving factoring quadratic expressions. It explains that the riddles involve two numbers whose product and sum are given. There are four possible combinations of signs for the product and sum. Examples are worked through step-by-step, showing how to determine the signs of the two numbers based on the information given, find the factors of the product, and write the factored quadratic expression that represents the solution. The goal is to help students recognize different patterns that arise when factoring quadratics and factor expressions into standard form.

1. Standard 2: Algebraic Thinking
2.4: Distinguish between linear and nonlinear functions through
informal investigations.
Objective: Riddle Quadratics into Quadratic Factored Form
Remember all the time spent factoring? Well your hard work will pay
off now because you need them in your head to factor quadratics
efficiently. (Never fear if you don't have them in your head, jotting them
down this week will refresh your memory and get you up to speed, but
more importantly, give you the correct answer.)
The examples below explain four different riddles you can get:
#1: Multiply to a positive and sum to a positive - both numbers positive
#2: Multiply to a positive and sum to a negative - both numbers
negative
#3 Multiply to a negative and sum to a negative- one number negative
(greatest absolute value) other positive
#4 Multiply to a negative and sum to a positive- one number positive the
other negative
"Riddle me this, riddle me that, who's afraid of the big black bat?"-- The
Riddler in Batman Forever
Riddle #1 I'm thinking of two numbers that Multiply to give me 8,
and Sum to give me 6.
Strategy: 1st: Figure out the sign of the two numbers.
The two numbers multiply to a positive number. This gives us two cases
to think about, however with the sum being positive, we can logically
find the correct path.
Case 1: Both the numbers are positive because "A positive times a
positive equals a positive."
A positive number added to a positive number equals a positive number.
Case 2: Both numbers are negative because "A negative times a negative
equals a positive."
A negative number added to a negative number equals a negative
number.
Logic: Since the numbers sum to a positive, case 2 is out and case 1:
both numbers positive.
2nd: Think of the factors of 8. (Jot them down if you ever get stuck)
Factors(8) = (1, 2, 4, 8)
Mentally sort through and find the pair that sum to 6. 1+8 not, 2+4
Yes!
2*4 = 8 and 2+4=6 + 2 an

2. Factor and Simplify
Quiz Practice
Instructions: Factor
i.e. x2 -2x - 8 = y
Answer: (x-4)(x+2) = y
1) x2 + 6x + 5 = y
2) x2 +12x + 32 = y
3) x2 + 3x + -54 = y
4) x2 + -2x +-24 = y
5) x2 -4x +4 = y
6) x2 + 14x 48 = y
7) x2 -4x +3 = y
8) x2 11x 18 = y
9) x2 -2x + -15 = y
10) x2 -1x -72 = y
11) x2+ 4x -32 = y
12) x2 -5x -14 = y
13) x2-9x + 8 = y
14) x2 + 5x -14 = y
15) x2 + 6x -27 = y
16) x2-3x -54 = y
17) x2+ 2x -15 = y
18) x2 + 9x + 8 = y
19) x2-1x -20 = y
20) x2-5x -6= y
Instructions: Simplify
i.e. (x +8)(x -3) = y
Answer: x2 + 5x-24 =y
21) (x -1)(x + 8) = y
22) (x + 4)(x -9) = y
23) (x + 9)(x -6) = y
24) (x + 48)(x -54) = y
25) (x + 3)(x + 8) = y
26) (x + 8)(x + -1) = y
27) (x -54)(x + 8) = y
28) (x -6)(x -54) = y
29) (x -6)(x + 8) = y
30) (x -4)(x + 6) = y
31) (x -4)(x + 9) = y
32) (x -5)(x + 3) = y
33) (x -4)(x + 8) = y
34) (x +12)(x + 4) = y
35) (x -2)(x -9) = y
36) (x-2)(x + 4) = y
3rd: We are working with quadratics so this answer must be put in what
is called a "factored quadratic" form. Factored Quadratic should be easy
to remember, you use your knowledge of "factors" to answer the riddle.
4th: Here is the Quadratic factored form:(x + __ ) (x + __ ) = y
Insert the answers + 2 and + 4 into the equation. Answer: (x + 4)(x
+ 2) = y
Riddle # 2 I'm thinking of two numbers that Multiply to give me 12,
and Sum to give me -7.
Strategy: 1st: Figure out the sign of the two numbers. See Case 2 above.
Both numbers must be negative to multiply to a positive and sum to a
negative.
Riddle # 3I'm thinking of two numbers that Multiply to give me -10, and
Sum to give me -3.
Strategy: 1st: Figure out the sign of the two numbers. The two numbers'
product is a negative number, so one of the numbers must be negative
and the other must be positive. " A negative times a positive equals a
negative."
Since the sum of the two numbers is negative, when we find the factors,
the greatest one will get a negative sign put on it so the numbers will, in
fact, sum to a negative.
2nd: Think of the factors of 10. (Jot them down if you ever get stuck)
Factors(10) = (1, 2, 5, 10)
Mentally sort through and find the pair that have a difference of 3. 10-1
not, 5-2 Yes!
3rd: Write down write down the factors that fit 2 5
4th: Figure out which needs the negative sign so they sum to -3.
2 - 5 = -3
2 and -5 are the answer to the riddle. Insert answers into the "Quadratic
Factored" form.
Answer: (x + 2)(x - 5)= y
Riddle #4 I'm thinking of two numbers that Multiply to give me -15, and
Sum to give me 2.
Strategy: 1st: Figure out the sign of the two numbers. One of the
numbers must be negative and the other must be positive. 2nd: Since the
sum of the two numbers is positive, when we find the factors the
greatest one will remain positive and the least will get a negative sign
put on it