Question

Find the exact solutions to ​f(x)equals=0 in the complex numbers and confirm that the solutions are not real by showing that the graph of yequals=​f(x) does not cross the​ x-axis.
xsquared2plus+8181equals=0
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Question content area right
Part 1
xequals=enter your response here
​(Simplify your answer. Type your answer in the form aplus+bii. Type an exact​ answer, using radicals as needed. Use a comma to separate answers as​ needed.)

Answer 1

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 \( y = f(x) = x^2 + 8181 \).

1. The function \( f(x) \) is a parabola that opens upwards (since the coefficient of \( x^2 \) is positive).
2. The vertex of the parabola can be found by noticing that the minimum value occurs when \( x = 0 \).
3. 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.