# Show that x is a square expression

1. Jun 16, 2011

### Shawn Garsed

1. The problem statement, all variables and given/known data
n(n+3)(n+6)(n+9)=x-81
Show that x is a squared expression

2. Relevant 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

3. 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.

2. Jun 16, 2011

### tiny-tim

Hi Shawn!

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 … )

3. Jun 16, 2011

### phinds

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. Jun 16, 2011

### phinds

Well, see, that's the problem with using a calculator. They keep you from learning how to THINK.

5. Jun 16, 2011

### tiny-tim

Shawn, that's obviously wrong!

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

Try again!

6. Jun 16, 2011

### Shawn Garsed

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. Jun 16, 2011

### tiny-tim

Shawn, it was nearly right …

try expanding (n2 + 9n + c)2

8. Jun 16, 2011

### Shawn Garsed

That really helped, I have the answer now.

((x+4.5)2-11.25)2

Thanks.

9. Jun 16, 2011

### tiny-tim

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

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

Last edited by a moderator: Apr 26, 2017