To solve the equation \( x^2 + 8181 = 0 \), we can rearrange it:
\[ x^2 = -8181 \]
Next, we take the square root of both sides to find the solutions:
\[ x = \pm \sqrt{-8181} \]
Since we are dealing with a negative number under the square root, we can express the solution in terms of imaginary numbers:
\[ x = \pm i \sqrt{8181} \]
Now, we need to simplify \( \sqrt{8181} \). To do this, we can factor \( 8181 \) to see if it has any perfect squares. The prime factorization of \( 8181 \) is:
\[ 8181 = 3 \times 2727 = 3 \times 3 \times 909 = 3^2 \times 3 \times 303 = 3^3 \times 3 \times 101 = 3^3 \times 101 \]
Thus, we can take the square root:
\[ \sqrt{8181} = 3\sqrt{909} \]
Since \( 909 = 9 \times 101 = 3^2 \times 101 \), we can write:
\[ \sqrt{909} = 3\sqrt{101} \]
So:
\[ \sqrt{8181} = 3 \cdot 3\sqrt{101} = 9\sqrt{101} \]
Therefore, the solutions can be rewritten as:
\[ x = \pm i \cdot 9\sqrt{101} \]
In summary, the solutions to \( f(x) = 0 \) are:
\[ x = 9i\sqrt{101}, -9i\sqrt{101} \]
Part 1: The exact solutions are \( 9i\sqrt{101}, -9i\sqrt{101} \).
To confirm that the solutions are not real, we can analyze the graph of \( y = f(x) = x^2 + 8181 \).
- The function \( f(x) \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive).
- The vertex of the parabola can be found by noticing that the minimum value occurs when \( x = 0 \).
- Calculate \( f(0) = 0^2 + 8181 = 8181 \).
Since the minimum value of \( f(x) \) is \( 8181 \), which is positive, the graph does not cross the x-axis. Therefore, there are no real solutions, confirming that both solutions are complex.
In conclusion, the solutions are \( 9i\sqrt{101}, -9i\sqrt{101} \) and the graph of \( f(x) \) does not cross the x-axis.