Show that x is a square expression

[Shawn Garsed]

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.

 

[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 … )

 

[phinds]

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 … )

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.

 

[phinds]

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.

 

[tiny-tim]

(n²+9n)²+n²+9n+4.5

Shawn, that's obviously wrong!

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

Try again! ​

 

[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.

 

[tiny-tim]

Shawn, it was nearly right …

try expanding (n² + 9n + c)² ​

 

[Shawn Garsed]

Shawn, it was nearly right …

try expanding (n² + 9n + c)² ​

That really helped, I have the answer now.

((x+4.5)²-11.25)²

Thanks.

 

[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 ​