Questions: Solve the following equation by making an appropriate substitution. x^-2+6 x^-1+5=0 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Use a comma to separate answers as needed.) B. The solution set is the empty set.

Solution Steps

We start with the equation

\[ x^{-2} + 6x^{-1} + 5 = 0. \]

To simplify this equation, we make the substitution

\[ y = x^{-1}. \]

This transforms the equation into

\[ y^2 + 6y + 5 = 0. \]

Next, we solve the quadratic equation

\[ y^2 + 6y + 5 = 0. \]

Factoring the quadratic, we find the solutions for \( y \):

\[ (y + 5)(y + 1) = 0. \]

Thus, the solutions are

\[ y = -5 \quad \text{and} \quad y = -1. \]

Now, we substitute back to find the values of \( x \) using \( y = x^{-1} \):

1. For \( y = -5 \): \[ x^{-1} = -5 \implies x = -\frac{1}{5}. \]

2. For \( y = -1 \): \[ x^{-1} = -1 \implies x = -1. \]

Final Answer