Find the number of real solutions to the system y = x^2 - 5, x^2 + y^2 = 25.
+1262 so we get y=4 and x=3since the only integer for x^2+y^2=25 is 3,4,5 since \(\sqrt{25} \)=5 and by the pythagorean theorem either x or y=3 or 4 but to be specific we must check and ony 4=3^2-5 works so x=3 , y=4
+9519 We can actually solve the equation to see how many real solutions there are.
\(\begin{cases}y = x^2 - 5\\x^2 + y^2 = 25\end{cases}\\ x^2 + (x^2 - 5)^2 = 25\\ x^4 - 9x^2 = 0\\ x^2 = 0 \text{ or } x^2 = 9\\ x = -3, 0, 3\)
Plugging in these values into the equation gives
\(y = 4, -5, 4\)
in this order.
Therefore there are 3 pairs of real solution, namely \((-3, 4), (0, -5), \text{ and }(3, 4)\)
