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A number has the property that its square is equal to 84 more than five times the number. Enter all numbers that have this property.

 May 20, 2020
 #4
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A number has the property that its square is equal to 84 more than five times the number. Enter all numbers that have this property. 

 

I don't know what AoPS is.  All I know is that you asked for help and I'll help you. 

 

The problem calls for a certain number.                    Let's let X represent that number. 

                                                                                  Let's let X2 represent its square. 

                                                                                  Let's let 5X represent five times the number.  ?

 

X2 is larger than 5X.  It's 84 more than 5X. 

So let's add 84 to 5X to make them balance. 

 

Set it up like this:                                                         X2  =  5X + 84  

 

Get all the terms on one side 

and then set it equal to zero.                                       X2 – 5X – 84  =  0 

 

Have you studied how to factor quadratic equations yet? 

In this instance, you have to find two numbers that multiply to 84 and add to 5. 

And, of course, you have to use the proper polarity to make it come out right. 

 

I'll do this one for you.                                                 (X – 12)(X + 7)  =  0 

 

As happens in many quadratics,

there are two answers.                                                 X = +12  &  X = –7 

.

 May 20, 2020

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