Question

solve the equation by making an appropriate substitution.
$x^4 - 170x^2 + 169 = 0$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the solution set is {}.
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. the solution set is $\varnothing$.

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the equation becomes:
$$u^2 - 170u + 169 = 0$$

Step2: Factor the quadratic equation

Factor the quadratic:
$$(u - 1)(u - 169) = 0$$

Step3: Solve for $u$

Set each factor to 0:
$u - 1 = 0 \implies u=1$; $u - 169 = 0 \implies u=169$

Step4: Substitute back $u=x^2$

For $u=1$: $x^2=1 \implies x=\pm1$
For $u=169$: $x^2=169 \implies x=\pm13$

Answer:

A. The solution set is $\{1, -1, 13, -13\}$