What are the solutions to the equation [tex]$x-\frac{7}{x}=6$[/tex]?

A. [tex]$x=-7 \text { and } x=1$[/tex]
B. [tex]$x=-6 \text { and } x=-1$[/tex]
C. [tex]$x=-1 \text { and } x=7$[/tex]
D. [tex]$x=1 \text { and } x=6$[/tex]

Multiply both sides of the equation by x to get rid of the fraction: x 2 − 7 = 6 x .
Rearrange the equation into the standard quadratic form: x 2 − 6 x − 7 = 0 .
Factor the quadratic equation: ( x − 7 ) ( x + 1 ) = 0 .
Solve for x : x = 7 or x = − 1 . The solutions are x = − 1 and x = 7 ​ .

Explanation

Problem Analysis We are given the equation x − x 7 ​ = 6 and asked to find its solutions.

Eliminating the Fraction To solve this equation, we first want to eliminate the fraction. We can do this by multiplying both sides of the equation by x . This gives us: x ( x − x 7 ​ ) = 6 x x 2 − 7 = 6 x

Rearranging into Quadratic Form Now, we want to rearrange the equation into a standard quadratic form, which is a x 2 + b x + c = 0 . Subtracting 6 x from both sides, we get: x 2 − 6 x − 7 = 0

Factoring the Quadratic Next, we factor the quadratic equation. We are looking for two numbers that multiply to − 7 and add to − 6 . These numbers are − 7 and 1 . So, we can factor the equation as: ( x − 7 ) ( x + 1 ) = 0

Solving for x To find the solutions for x , we set each factor equal to zero: x − 7 = 0 or x + 1 = 0

Final Solutions Solving these equations, we get: x = 7 or x = − 1 Thus, the solutions to the equation are x = 7 and x = − 1 .


Examples
Quadratic equations like this one appear in many real-world situations, such as calculating the trajectory of a ball, determining the dimensions of a garden to enclose a certain area, or even in financial models to predict growth and decay. Understanding how to solve them allows us to make informed decisions and predictions in these scenarios.

Answered by GinnyAnswer | 2025-07-03

The solutions to the equation x − x 7 ​ = 6 are x = 7 and x = − 1 . Therefore, the correct option is C: x = − 1 and x = 7 .
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Answered by Anonymous | 2025-07-04