## 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)Adding these gives

\[x^2-xy+y^2+xy=50 \rightarrow x^2+2xy+y^2=50\]

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.

\[x+y=\pm5 \sqrt{2}\]

\[x-y=\pm\sqrt{2}\]

Adding these gives

\[2x=\pm 6 \sqrt{2} \rightarrow x = \pm 3 \sqrt{5}\]

.Subtracting these gives

\[2y=\pm 4 \sqrt{2} \rightarrow y = \pm 2 \sqrt{5}\]

.