If x and p are both greater than zero and 4x^2p^2+xp-33=0, then what is the value of p in terms of x?

A) - 3/x

B) - 11/4x

C) 3/4x

D) 11/4x

11 / (4x)

Explanation:

1) Make a change of variable:

u = xp

2) The new equation with u is:

4x²p² + xp - 33 = 0 4 (xp) ² + xp - 33 = 0 4u² + u - 33 = 0

3) Factor the left side of the new equation:

Split u as 12u - 11u ⇒ 4u² + u - 33 = 4u² + 12u - 11u - 33 Group terms: (4u² + 12u) - (11u + 33) Extract common factor of each group: 4u (u + 3 - 11 (u + 3) Common factor u + 3: (u + 3) (4u - 11).

4) Come back to the equation replacing the left side with its factored form and solve:

(u + 3) (4u - 11) = 0 Use zero product propery: u + 3 = 0 or 4u - 11 = 0 solve each factor: u = - 3 or u = 11/4

5) Come back to the original substitution:

u = xp

If u = - 3 ⇒ xp = - 3 ⇒ x or p is negative and that is against the condition that x and p are both greater than zero, so this solution is discarded.

Then use the second solution:

u = xp = 11/4

Solve for p:

Divide both sides by x: p = 11 / (4x), which is the option D) if you write it correctly.