Solve by Factoring p^4-625=0

Problem

p4−625=0

Solution

1. Identify the expression as a difference of squares, since p4=(p2)2 and 625=25

2. Apply the formula for the difference of squares, a2−b2=(a−b)*(a+b) to the left side of the equation.

(p2−25)*(p2+25)=0

3. Identify that the first factor p2−25 is also a difference of squares, where 25=5

4. Factor again using the difference of squares formula for the term (p2−25)

(p−5)*(p+5)*(p2+25)=0

5. Set each factor to zero using the zero product property to find the roots.

p−5=0⇒p=5

p+5=0⇒p=−5

p2+25=0⇒p2=−25

6. Solve for p in the quadratic factor by taking the square root of both sides, noting that the square root of a negative number involves the imaginary unit i

p=±√(,−25)

p=±5*i

Final Answer

p=5,−5,5*i,−5*i

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