Question

what are the solutions to the equation $x - \frac{7}{x} = 6$?\
$x = 1$ and $x = 6$\
$x = -1$ and $x = 7$\
$x = -6$ and $x = -1$\
$x = -7$ and $x = 1$

Explanation:

Step1: Transform the equation

Multiply both sides of the equation \( x - \frac{7}{x}=6 \) by \( x \) (assuming \( x
eq0 \)) to get rid of the fraction: \( x^2 - 7 = 6x \).

Step2: Rearrange into standard quadratic form

Rearrange the equation to \( x^2 - 6x - 7 = 0 \).

Step3: Factor the quadratic equation

Factor the quadratic: \( (x - 7)(x + 1)=0 \).

Step4: Solve for x

Set each factor equal to zero: \( x - 7 = 0 \) gives \( x = 7 \), and \( x + 1 = 0 \) gives \( x=-1 \).

Answer:

\( x = -1 \) and \( x = 7 \)