what is the value of square Z over the integers in general case?

Question

if i have $4z^2-4(3x-5)^2 = (3x-5+45y)^2-(3x-5-27y)^2$

and i am trying to get short expression for $z^2$

can you help me to reduce the expression to get short expression for the value of $z^2$ ?

Thanks,

Answer 1

You have a difference of squares that you can iron out on the right side. When you factor it out and move $4(3x-5)^2$ to the RHS, you get $$4z^2 = 4(3x-5)^2+(6x-10+18y)(72y)$$

And divide by 4 on each side to get $$z^2 = (3x-5)^2+(6x-10+18y)(18y)$$

If you want to expand a little more, feel free. You'll get

$$z^2 = 9x^2 -30x+25 + 108xy - 180y + 324y^2 \\= 9x^2 + 108xy + 324y^2 -30x-180y$$

Answer 2

Thank you Meow now i can see that $z^2=(3x−5)^2+(6x−10+18y)(18y)$

so if we let, $3x-5=v ,18y=u$ ,

then $z^2=v^2+(2v+u)(u)$

so $z^2=(v+u)^2$ which mean , $z=18y+3x-5$