Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

solve

where x
and y
denote real numbers.

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 164

Re: solve

One solution is:
x=1802
y=1860

KerimF

Member

From: Aleppo-Syria

Registered: 2018-08-10

Posts: 164

Re: solve

From the original equation:
[1] x-1801+sqrt[(y-1860)*(x-1801)] = 2*sqrt(x-1801) - 1
[2] sqrt[(y-1860)*(x-1801)] = -(x-1801)+ 2*sqrt(x-1801) - 1
[3] sqrt[(y-1860)*(x-1801)] = -[sqrt(x-1801) - 1]^2

Also from the original equation:
x-1801 > 0
x > 1801

From the equation [3] above:
The right side is always negative.
And for the sqrt on left side to exist, the right side should be 0.
[4] [sqrt(x-1801) - 1] = 0
[5] sqrt(x-1801) = 1
[5] (x-1801) = 1
[6] x = 1801 + 1
[7] x = 1802

Also the left side of [3] should be zero.
[8] sqrt[(y-1860)*(x-1801)] = 0
[9] (y-1860)*(x-1801) = 0
By replacing x = 1802
[10] (y-1860)* 1 = 0
[11] (y-1860) = 0
[12] y = 1860

Therefore, there is one solution only:
x = 1802
y = 1860