Show that x is a square expression

In summary, Shawn found that n(n+3)(n+6)(n+9)+81 is a squared expression, and that using a graphing calculator doesn't help. He then tried expanding it, but got an expression that is wrong. He then tried again, and realized that it was wrong. He then used his other hint to find that x2 + ax + b is the correct answer.
  • #1
Shawn Garsed
50
0

Homework Statement


n(n+3)(n+6)(n+9)=x-81
Show that x is a squared expression


Homework Equations


These are some examples they give you:
0*3*6*9=92-81
1*4*7*10=192-81
2*5*8*11=312-81


The Attempt at a Solution


To be honest, I have know idea where to start. So, I'd like a push in the right direction.
 
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  • #2
Hi Shawn! :smile:

You need to show that n(n+3)(n+6)(n+9) + 81 is a square …

have you tried expanding it?​

(alternatively, you could try finding a quadratic formula that fits 9,19,31 … :wink:)
 
  • #3
tiny-tim said:
Hi Shawn! :smile:

You need to show that n(n+3)(n+6)(n+9) + 81 is a square …

have you tried expanding it?​

(alternatively, you could try finding a quadratic formula that fits 9,19,31 … :wink:)

Very nice, Tiny-tim. I'm an old guy who remembers very little math but I love math problems, so try to work them out. This one had me flummoxed until I read your first hint and then after recovering from the headache caused by smacking myself in the forehead so hard I saw that it pretty much drops right out.
 
  • #4
Shawn Garsed said:
I've tried expanding it, but all I got from that (using a graphing calculator) is this expression: (n2+9n)2+n2+9n+4.5. But I don't know where to go from there.

Well, see, that's the problem with using a calculator. They keep you from learning how to THINK.
 
  • #5
Shawn Garsed said:
(n2+9n)2+n2+9n+4.5

Shawn, that's obviously wrong! :rolleyes:

(how can it have a ".5" ?? :wink:)

Try again! :smile:
 
  • #6
I just realized it's wrong, I deleted the reply. I expanded the expression, but I don't know where to go from there, I used a graphing calculator to graph the expression, it looks like a quadratic graph with the negative y-values mirrored over the x-axis, which makes sense cause the expression equals a square which is always positive.
 
  • #7
Shawn, it was nearly right …

try expanding (n2 + 9n + c)2 :wink:
 
  • #8
tiny-tim said:
Shawn, it was nearly right …

try expanding (n2 + 9n + c)2 :wink:

That really helped, I have the answer now.

((x+4.5)2-11.25)2

Thanks.
 
  • #9
For future reference, the way to square-root a polynomial is to start from the left, and work your way across (a bit like long division) …

for x4 + ax3 + bx2 + cx + d,

https://www.physicsforums.com/library.php?do=view_item&itemid=107" for the x4 + ax3 part, and then add-on a unit to complete the square for the whole thing :wink:

btw, my other hint was …

9 19 31

10 12

2

the 2 means it must be x2 + ax + b,

subtracting 02 12 and 22 from 9 19 and 31 gives

9 18 and 27, so a = b = 9 :biggrin:
 
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FAQ: Show that x is a square expression

1. What does it mean to show that x is a square expression?

To show that x is a square expression means to prove that it can be written in the form x = a^2, where a is a constant or a variable. In other words, x is a perfect square.

2. How do you show that x is a square expression?

To show that x is a square expression, you can use various methods such as completing the square, factoring, or using the quadratic formula. These methods involve manipulating the expression until it is in the form x = a^2. If you are able to do this, then you have successfully shown that x is a square expression.

3. Why is it important to show that x is a square expression?

Showing that x is a square expression can help in solving equations and simplifying expressions. It also allows us to better understand the properties and relationships of quadratic functions, which are commonly used in many areas of science and mathematics.

4. Can x be a negative number and still be a square expression?

No, x cannot be a negative number and still be a square expression. This is because a square expression is defined as a number that can be written as the square of a real number. Since the square of any real number is always positive, x must also be positive.

5. Is there a difference between showing that x is a square expression and proving that x is a square expression?

No, there is no difference between showing and proving that x is a square expression. Both terms refer to the same process of demonstrating that an expression can be written in the form x = a^2, where a is a constant or a variable. The term "showing" is often used in more informal contexts, while "proving" is commonly used in formal settings such as in mathematics research.

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