Question

solve.
\\(\left(x^{2} - 17\
ight)^{2} - 16\left(x^{2} - 17\
ight) + 64 = 0\\)
if there is more than one solution, separate them with commas.

Explanation:

Step1: Let \( y = x^2 - 17 \), then the equation becomes \( y^2 - 16y + 64 = 0 \)

This is a substitution to simplify the quadratic in terms of \( y \).

Step2: Factor the quadratic equation \( y^2 - 16y + 64 = 0 \)

We know that \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( a = y \), \( 2ab = 16y \) so \( b = 8 \). Thus, \( y^2 - 16y + 64=(y - 8)^2 = 0 \)

Step3: Solve for \( y \)

From \( (y - 8)^2 = 0 \), we get \( y - 8 = 0 \), so \( y = 8 \)

Step4: Substitute back \( y = x^2 - 17 \) and solve for \( x \)

We have \( x^2 - 17 = 8 \), then \( x^2=8 + 17=25 \), so \( x=\pm\sqrt{25}=\pm5 \)

Answer:

-5, 5