4x^2-9y+32x-144y-548=0
4x^2+32x+64 -548+64=144y
4(x+4)^2-484=144y
(x+4)^2-121=38y
the left is a difference of two squares...
(x+4-11)(x+4+11)=38y
check all that I did it in my head.
I'd have stopped a step earlier:
y = (x+4)^2/38 - 121/38
The given equation is 4x^2 - 9y + 32x - 144y - 548 = 0. To find any possible solution(s), we can simplify the equation and try to solve for x and y.
First, let's group the x terms and the y terms together:
4x^2 + 32x - 9y - 144y - 548 = 0
Rearrange the terms:
4x^2 + 32x - 153y - 548 = 0
Now, let's focus on the x terms. We can factor out 4 from the first two terms:
4(x^2 + 8x) - 153y - 548 = 0
To complete the square for the x terms, we need to add half of the coefficient of x (8) squared. That would be (8/2)^2 = 16.
4(x^2 + 8x + 16) - 16(4) - 153y - 548 = 0
Simplify:
4(x + 4)^2 - 64 - 153y - 548 = 0
Combine like terms:
4(x + 4)^2 - 153y - 612 = 0
Now let's move on to the y terms. We can factor out -153:
4(x + 4)^2 - 153(y + 4) - 612 = 0
To complete the square for the y terms, we need to add half of the coefficient of y (-153) squared. That would be (-153/2)^2 = 23409/4.
4(x + 4)^2 - 153(y + 4 + 23409/4) - 612 = 0
Simplify:
4(x + 4)^2 - 153(y + 4 + 5852.25) - 612 = 0
Combine like terms:
4(x + 4)^2 - 153(y + 5856.25) - 612 = 0
Let's rearrange the terms again to separate x and y:
4(x + 4)^2 - 153(y + 5856.25) = 612
Now we want to isolate (x + 4)^2:
4(x + 4)^2 = 153(y + 5856.25) + 612
Divide both sides by 4:
(x + 4)^2 = (153/4)(y + 5856.25) + 153
Take the square root of both sides:
x + 4 = sqrt[(153/4)(y + 5856.25) + 153]
Subtract 4 from both sides:
x = sqrt[(153/4)(y + 5856.25) + 153] - 4
Therefore, the equation 4x^2 - 9y + 32x - 144y - 548 = 0 simplifies to x = sqrt[(153/4)(y + 5856.25) + 153] - 4.