Find the exact solutions to ​f(x)equals0 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.

Question

x squared plus 10 x plus 34 equals 0 nbsp
Question content area bottom
Part 1
The​ solution(s) is/are xequals
  
negative 5 plus 3 i.
​(Simplify your answer. Type your answer in the form aplusbi. Type an exact​ answer, using radicals as needed. Use a comma to separate answers as​ needed.)

Answer 1

To solve the equation \(x^2 + 10x + 34 = 0\) in the complex numbers, we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this equation, \(a = 1\), \(b = 10\), and \(c = 34\).

First, we calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 - 4ac = 10^2 - 4 \cdot 1 \cdot 34 = 100 - 136 = -36 \]

Since the discriminant is negative, we will have complex solutions.

Now, we will find the solutions using the quadratic formula:

\[ x = \frac{-10 \pm \sqrt{-36}}{2 \cdot 1} \]

Calculating the square root of \(-36\):

\[ \sqrt{-36} = 6i \]

Substituting this back into the quadratic formula gives:

\[ x = \frac{-10 \pm 6i}{2} \]

Now we can simplify this:

\[ x = \frac{-10}{2} \pm \frac{6i}{2} \] \[ x = -5 \pm 3i \]

Thus, the exact solutions are:

\[ x = -5 + 3i, \quad x = -5 - 3i \]

Part 2: Confirming the solutions are not real by showing the graph does not cross the x-axis.

To confirm that the solutions are not real, we can analyze the quadratic function \(f(x) = x^2 + 10x + 34\).

The vertex of the parabola represented by this equation can be found using the vertex formula \(x = -\frac{b}{2a}\):

\[ x = -\frac{10}{2 \cdot 1} = -5 \]

Now, we substitute \(x = -5\) back into \(f(x)\) to find the value of the function at the vertex:

\[ f(-5) = (-5)^2 + 10(-5) + 34 \] \[ = 25 - 50 + 34 \] \[ = 9 \]

Since \(f(-5) = 9\) and is positive, the vertex is above the x-axis, indicating that the parabola does not cross the x-axis.

Overall, the solutions are confirmed as:

\[ x = -5 + 3i, \quad x = -5 - 3i \]

Therefore, the final answer is:

\[ \text{x equals } -5 + 3i, -5 - 3i \]