Question

solve the equation by using substitution. express numbers in simplified form using integers, fractions, or radicals.\\(\left(p^{2}-2 p\
ight)^{2}-2\left(p^{2}-2 p\
ight)=3\\)\
the solution set is \\(\square\\).

Explanation:

Step1: Let \( u = p^2 - 2p \)

Substitute \( u \) into the equation, we get \( u^2 - 2u = 3 \).

Step2: Rearrange to standard quadratic form

Subtract 3 from both sides: \( u^2 - 2u - 3 = 0 \).

Step3: Factor the quadratic equation

Factor \( u^2 - 2u - 3 \) as \( (u - 3)(u + 1) = 0 \).

Step4: Solve for \( u \)

Set each factor equal to zero: \( u - 3 = 0 \) or \( u + 1 = 0 \), so \( u = 3 \) or \( u = -1 \).

Step5: Substitute back \( u = p^2 - 2p \)

Case 1: \( p^2 - 2p = 3 \)

Rearrange to \( p^2 - 2p - 3 = 0 \), factor as \( (p - 3)(p + 1) = 0 \), so \( p = 3 \) or \( p = -1 \).

Case 2: \( p^2 - 2p = -1 \)

Rearrange to \( p^2 - 2p + 1 = 0 \), factor as \( (p - 1)^2 = 0 \), so \( p = 1 \) (double root).

Answer:

\(\{-1, 1, 3\}\)