We are asked to solve the system of equations: $x^2 + y^2 = 3$ $x^2 - 15 = y$

1. Problem Description

We are asked to solve the system of equations:

$x^2 + y^2 = 3$

$x^2 - 15 = y$

2. Solution Steps

We have the system of equations:

$x^2 + y^2 = 3$ (1)

$x^2 - 15 = y$ (2)

From equation (2), we have $x^2 = y + 15$. Substituting this into equation (1), we get:

$(y + 15) + y^2 = 3$

$y^2 + y + 15 - 3 = 0$

$y^2 + y + 12 = 0$

We solve this quadratic equation for $y$. The discriminant is:

$\Delta = b^2 - 4ac = (1)^2 - 4(1)(12) = 1 - 48 = -47$

Since the discriminant is negative, there are no real solutions for $y$. Therefore, there are no real solutions for $x$.

However, let us proceed to find complex solutions.

Using the quadratic formula,

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{-47}}{2} = \frac{-1 \pm i\sqrt{47}}{2}$

We have two possible values for $y$:

$y_1 = \frac{-1 + i\sqrt{47}}{2}$ and $y_2 = \frac{-1 - i\sqrt{47}}{2}$

Now we find $x^2$ for each value of $y$:

$x^2 = y + 15$

For $y_1$, $x^2 = \frac{-1 + i\sqrt{47}}{2} + 15 = \frac{-1 + 30 + i\sqrt{47}}{2} = \frac{29 + i\sqrt{47}}{2}$

For $y_2$, $x^2 = \frac{-1 - i\sqrt{47}}{2} + 15 = \frac{-1 + 30 - i\sqrt{47}}{2} = \frac{29 - i\sqrt{47}}{2}$

Now, we would have to take the square root of these complex numbers to find the values of $x$, which is not trivial and might not be the intent of the problem. Since the problem does not specify that we need complex roots, it can be assumed that there are no real solutions.

3. Final Answer

No real solutions.