Question

X²+Y²=250 and XY=117
What are the values of X and Y?

Answer 1

x = -9, y = -13

or x = 13, y = 9

or x = -13, y = -9

or x = 9, y = 13

Explanation:

`x^2+y^2=250\\\\x^2-2xy+y^2+2xy=250\\\\(x-y)^2=250-2xy\\\\(x-y)^2=250-2\cdot117\\\\ (x-y)^2=16\\\\x-y=4\qquad\qquad\vee\qquad \qquad x-y=-4\\\\x=4+y \qquad\qquad \vee\qquad\qquad x=-4+y\\\\(y+4)y=117\qquad\vee\qquad\quad (y-4)y=117\\\\y^2+4y-117=0\qquad\vee\qquad y^2-4y-117=0\\\\y=(-4\pm√(4^2-4(-117)))/(2\cdot1)\qquad\vee\qquad y=(4\pm√(4^2-4(-117)))/(2\cdot1)\\\\y=(-4\pm√(16+468))/(2)\qquad\ \ \vee\qquad y=(4\pm√(16+468))/(2)`

`y_1=(-4-22)/(2)\ ,\quad y_2=(-4+22)/(2)\ ,\quad y_3=(4-22)/(2)\ ,\quad y_4=(4+22)/(2)\\\\y_1=-13\ ,\qquad y_2=9\ ,\qquad\quad\qquad\ y_3=-9\ ,\qquad y_4=13\\\\x_(1,2)=4+y_(1,2)\qquad\qquad\qquad\qquad\qquad x_(3,4)=-4+y_(3,4)\\\\x_1=-9\ ,\qquad x_2=13\ ,\qquad\quad\qquad x_3=-13\ ,\qquad x_4=9`