Solve the simultaneous equations $x + y = 2$ $x^2 + 2y = 12.$ - Edexcel - A-Level Maths Pure - Question 7 - 2005 - Paper 2

Now, substitute $y$ from the first equation into the second equation:

$x^2 + 2(2 - x) = 12.$

Expanding the equation gives:

$x^2 + 4 - 2x = 12.$
Subtracting 12 from both sides yields:

$x^2 - 2x - 8 = 0.$

Next, factor the equation:

$(x - 4)(x + 2) = 0.$
Thus, the solutions for $x$ are:

$x = 4 \quad \text{or} \quad x = -2.$

Using the values of $x$, substitute back to find corresponding $y$ values:

1. For $x = 4$: $y = 2 - 4 = -2.$
2. For $x = -2$: $y = 2 - (-2) = 4.$

The solutions to the simultaneous equations are:

1. $(x, y) = (4, -2)$
2. $(x, y) = (-2, 4)$.