You have the system of equations:

1. $x^2 + y^2 = 25$
2. $xy = 12$

To solve this system, we can use the identity $(x + y)^2 = x^2 + y^2 + 2xy$, which connects both equations.

Step 1: Use the identity $(x + y)^2 = x^2 + y^2 + 2xy$

Substitute the known values of $x^2 + y^2$ and $xy$ into this identity:

$(x + y)^2 = 25 + 2(12)$ $(x + y)^2 = 25 + 24$ $(x + y)^2 = 49$ $x + y = \pm 7$

So, $x + y = 7$ or $x + y = -7$.

Step 2: Solve for $x$ and $y$

Now, you have two possible scenarios for $x + y$:

• Case 1: $x + y = 7$
• Case 2: $x + y = -7$

For both cases, you can use the system of two equations:

1. $x + y = \pm 7$
2. $xy = 12$

Step 3: Solve the quadratic equations

Substitute $x + y = 7$ or $x + y = -7$ into the quadratic equation $t^2 - (x + y)t + xy = 0$, which simplifies to: $t^2 - (x + y)t + 12 = 0$

Case 1: $x + y = 7$

$t^2 - 7t + 12 = 0$

Solve this quadratic equation using the quadratic formula: $t = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(12)}}{2(1)}$ $t = \frac{7 \pm \sqrt{49 - 48}}{2}$ $t = \frac{7 \pm 1}{2}$

So, the solutions for $t$ are: $t = 4 \quad \text{or} \quad t = 3$

Thus, $x = 4$ and $y = 3$, or $x = 3$ and $y = 4$.

Case 2: $x + y = -7$

$t^2 + 7t + 12 = 0$

Solve this quadratic equation using the quadratic formula: $t = \frac{-7 \pm \sqrt{7^2 - 4(1)(12)}}{2(1)}$ $t = \frac{-7 \pm \sqrt{49 - 48}}{2}$ $t = \frac{-7 \pm 1}{2}$

So, the solutions for $t$ are: $t = -3 \quad \text{or} \quad t = -4$

Thus, $x = -3$ and $y = -4$, or $x = -4$ and $y = -3$.

Final Answer:

The possible solutions for $x$ and $y$ are:

• $(x, y) = (4, 3)$ or $(3, 4)$
• $(x, y) = (-3, -4)$ or $(-4, -3)$

Would you like to explore further details or have any questions? Here are some related questions to deepen your understanding:

1. How would the approach change if $x^2 + y^2 = 16$ and $xy = 6$?
2. What happens if one of the variables is restricted to be positive or negative?
3. How does solving for $(x + y)^2$ help in more complex systems?
4. Could this method be applied to higher-degree equations or more variables?
5. What are the geometric interpretations of these equations in the coordinate plane?

Tip: When dealing with quadratic systems like this, always check if you can apply identities (like $(x + y)^2$) to simplify and solve the system more easily.