Question Find all complex solutions to the equation $x + 2 + \frac{5}{x} = 0$ .

Studdy Solution

STEP 1

Assumptions
1. We are solving the equation $x + 2 + \frac{5}{x} = 0$ .
2. Solutions may be real or complex numbers.

STEP 2

To solve the equation, we need to find a common denominator for the terms involving $x$ . In this case, the common denominator is $x$ .
$x(x + 2) + 5 = 0$

STEP 3

Now, distribute the $x$ across the terms in the parentheses.
$x^2 + 2x + 5 = 0$

STEP 4

We have a quadratic equation in the form $ax^2 + bx + c = 0$ . We can solve for $x$ using the quadratic formula:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

STEP 5

Identify the coefficients $a$ , $b$ , and $c$ from the quadratic equation $x^2 + 2x + 5 = 0$ .
$a = 1, b = 2, c = 5$

STEP 6

Substitute the coefficients into the quadratic formula.
$x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

STEP 7

Simplify the expression inside the square root.
$x = \frac{-2 \pm \sqrt{4 - 20}}{2}$

STEP 8

Calculate the value inside the square root.
$x = \frac{-2 \pm \sqrt{-16}}{2}$

STEP 9

Since the value inside the square root is negative, we will have complex solutions. Recall that $\sqrt{-1} = i$ .
$x = \frac{-2 \pm 4i}{2}$

STEP 10

Divide both terms in the numerator by the denominator.
$x = -1 \pm 2i$

STEP 11

Write the solutions as a list of complex numbers.
The solutions are $-1 + 2i, -1 - 2i$ .

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