Question $x+\frac{5}{x}=4, x \neq 0$

Studdy Solution

STEP 1

1. The equation $x + \frac{5}{x} = 4$ involves a variable $x$ that is non-zero.
2. The goal is to solve for $x$ .
3. We will use algebraic manipulations to simplify and solve the given equation.

STEP 2

1. Eliminate the fraction by multiplying through by $x$ .
2. Rearrange the resulting quadratic equation.
3. Solve the quadratic equation using the quadratic formula.
4. Verify the solutions.

STEP 3

Multiply both sides of the equation by $x$ to eliminate the fraction:
$x \cdot x + x \cdot \frac{5}{x} = 4 \cdot x$

STEP 4

Simplify the resulting equation:
$x^2 + 5 = 4x$

STEP 5

Rearrange the equation to form a standard quadratic equation:
$x^2 - 4x + 5 = 0$

STEP 6

Solve the quadratic equation using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$ , $b=-4$ , and $c=5$ .
$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

STEP 7

Simplify inside the square root:
$x = \frac{4 \pm \sqrt{16 - 20}}{2}$

STEP 8

Continue simplifying:
$x = \frac{4 \pm \sqrt{-4}}{2}$

STEP 9

Since $\sqrt{-4} = 2i$ , where $i$ is the imaginary unit:
$x = \frac{4 \pm 2i}{2}$

STEP 10

Simplify further:
$x = 2 \pm i$

STEP 11

Verify the solutions by substituting back into the original equation. Let's check $x = 2 + i$ first:
$2 + i + \frac{5}{2 + i} = 4$

STEP 12

Multiply numerator and denominator by the conjugate:
$\frac{5}{2 + i} \cdot \frac{2 - i}{2 - i} = \frac{5(2 - i)}{(2 + i)(2 - i)} = \frac{10 - 5i}{4 + 1} = \frac{10 - 5i}{5} = 2 - i$

STEP 13

Add the simplified form back to the original equation:
$2 + i + 2 - i = 4$
$4 = 4$
Hence, $x = 2 + i$ is a solution.

STEP 14

Now, check $x = 2 - i$ :
$2 - i + \frac{5}{2 - i} = 4$

STEP 15

Multiply numerator and denominator by the conjugate:
$\frac{5}{2 - i} \cdot \frac{2 + i}{2 + i} = \frac{5(2 + i)}{(2 - i)(2 + i)} = \frac{10 + 5i}{4 + 1} = \frac{10 + 5i}{5} = 2 + i$

STEP 16

Add the simplified form back to the original equation:
$2 - i + 2 + i = 4$
$4 = 4$
Hence, $x = 2 - i$ is also a solution.
Solution: The solutions to the equation $x + \frac{5}{x} = 4$ are $x = 2 + i$ and $x = 2 - i$ .

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