Question Find the number $x$ such that $x + 5/x = 6$ .

Studdy Solution

STEP 1

Assumptions
1. Let the number be denoted as $x$ .
2. The reciprocal of $x$ is $\frac{1}{x}$ .
3. The equation to solve is $x + 5 \left( \frac{1}{x} \right) = 6$ .

STEP 2

Write down the equation based on the given information.
$x + 5\left(\frac{1}{x}\right) = 6$

STEP 3

To solve for $x$ , we need to clear the fraction by multiplying both sides of the equation by $x$ .
$x^2 + 5 = 6x$

STEP 4

Rearrange the equation to set it to zero, which is the standard form for a quadratic equation.
$x^2 - 6x + 5 = 0$

STEP 5

Factor the quadratic equation, if possible.
$(x - 1)(x - 5) = 0$

STEP 6

Apply the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero.
$x - 1 = 0 \quad \text{or} \quad x - 5 = 0$

STEP 7

Solve each equation for $x$ .
For $x - 1 = 0$ :
$x = 1$
For $x - 5 = 0$ :
$x = 5$

STEP 8

Verify that both solutions satisfy the original equation.
For $x = 1$ :
$1 + 5\left(\frac{1}{1}\right) = 1 + 5 = 6$
For $x = 5$ :
$5 + 5\left(\frac{1}{5}\right) = 5 + 1 = 6$
Both solutions satisfy the original equation.

STEP 9

State the final solution.
The numbers that satisfy the given equation are $x = 1$ and $x = 5$ .

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