Question

For what non-zero value of x,


`p^4 - 2p^2x^2 + x^4 + x`
will be a perfect square?

`a) (1)/(4p) \\ b)4p \\ c) {4p}^(2) \\ d) \frac{1}{ {4p}^(2) }`
I need step-by-step explanation. No spam. Thank you in anticipation.

Answer 1

• d) 1/(4p²)

Explanation:

Let's complete the square first:

• p⁴ - 2p²x² + x⁴ + x =
• p⁴ + 2p²x² + x⁴ - 4p²x² + x =
• (p² + x²)² - (4p²x² - x)

In order the expression above to be a perfect square, we need:

• 4p²x² - x = 0

Solve it for x:

• 4p²x² = x
• 4p²x = 1
• x = 1/(4p²)

Correct choice is d