# Quadratic question with two pairs of solutions

#### Rich16

##### New member
"The product of two numbers is 360. If one number is 16 less than double the other number, find the two possible number pairs."

This is my working:

x(2x-16)= 360
2x2-16x= 360
2x2-16x-360= 0
2(x2-8x-180)= 0
2(x-18)(x+10)= 0
x = 18 or -10

First pair: 18 and 10 (changing the ten from negative to positive)

I thought the second pair was just -18 and -10 but the textbook says -10 and -36. The quadratic formula of course just returns the same result as the factorisation above. I'm not sure how to arrive at the textbook's answer.

#### lev888

##### Full Member
How can the first pair be 18 and 10 if their product is not 360?
What does "changing the ten from negative to positive" mean?

#### Romsek

##### Full Member
$$\displaystyle x(2x-16) = 360\\ 2x^2 - 16x - 360 = 0\\ x^2 - 8x - 180 = 0\\ (x-18)(x+10) = 0\\ x=18,~-10\\~\\ x=18 \Rightarrow (2x-16) = 20\\ x = -10 \Rightarrow (2x-16) = -36\\~\\ \text{The pairs are (18,20) and (-10,-36)}$$

#### Dr.Peterson

##### Elite Member
It's important to define your variable: your x is the "other number" in the pair. It is not both numbers.

So when you get two values for x, they are not the pair; they are two possible values for one of the numbers, from each of which you can find another number to complete a pair, by evaluating 2x-16, or by dividing 360 by x.

#### Jomo

##### Elite Member
"The product of two numbers is 360. If one number is 16 less than double the other number, find the two possible number pairs."

This is my working:

x(2x-16)= 360
2x2-16x= 360
2x2-16x-360= 0
2(x2-8x-180)= 0
2(x-18)(x+10)= 0
x = 18 or -10

First pair: 18 and 10 (changing the ten from negative to positive)

I thought the second pair was just -18 and -10 but the textbook says -10 and -36. The quadratic formula of course just returns the same result as the factorisation above. I'm not sure how to arrive at the textbook's answer.
If you get x=18 and x=-10 (and think that is the answer) then why would you change your answer to something else, like x=18 and x=10. You must understand that 10 is as different from -10 as much as 15 is different from -10. Just because they look similar does not mean they are equal or interchangeable!

Also, as Dr Peterson stated, you MUST define your variables. One number is x and the other number is 2x-16.

Now lets look at your first equation. It says that x(2x-16)= 360. It DOES NOT SAY x*x=360 and this is what you are (somewhat) trying to say. Most importantly even if x*x=360 is the correct equation, you can't plug in different values for x! You can NOT replace one x with 18 and the other x with 10 or -10. NO!