Quadratic question with two pairs of solutions

Rich16

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Sep 20, 2019
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"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

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Jan 16, 2018
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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

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Nov 16, 2013
Messages
784
\(\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

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Nov 12, 2017
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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

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Joined
Dec 30, 2014
Messages
5,407
"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!
 
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