## Quadratic Simultaneous Equations With Mixed and Quadratic Terms in Both Equations

Simultaneous Equations with square and mixed
$xy$
terms in both equations can be quite tricky to solve.
$x^2+xy=30$
(1)
$y^2+xy=20$
(1)
$x^2-xy+y^2+xy=50 \rightarrow x^2+2xy+y^2=50$

We can factorise as
$(x+y)^2=50 \rightarrow x+y=\pm \sqrt{50}=\pm5 \sqrt{2}$
(3)
(1)-(2) gives
$x^2-y^2=10 \rightarrow (x+y)(x-y)=10$
(4)
(4) divided by (3) gives
$x-y=\pm \sqrt{2}$
(5)
Now we have ordinary simultaneous equations.
$x+y=\pm5 \sqrt{2}$

$x-y=\pm\sqrt{2}$

$2x=\pm 6 \sqrt{2} \rightarrow x = \pm 3 \sqrt{5}$
$2y=\pm 4 \sqrt{2} \rightarrow y = \pm 2 \sqrt{5}$