CAMANISHPARMAR wrote:

If the sum of the squares of x and y is 3, and \(x^4 = y^4 + 25\), what is the value of \(x^2\)?

A) \(\frac{-8}{3}\)

B) \(\frac{14}{3}\)

C) \(\frac{17}{3}\)

D) \(\frac{28}{3}\)

E) \(\frac{34}{3}\)

We are given that x^2 + y^2 = 3 and x^4 = y^4 + 25.

Let’s simplify the second equation:

x^4 = y^4 + 25

x^4 - y^4 = 25

(x^2 + y^2)(x^2 - y^2) = 25

Since x^2 + y^2 = 3, we have:

3(x^2 - y^2) = 25

x^2 - y^2 = 25/3

Now, adding x^2 - y^2 = 25/3 and x^2 + y^2 = 3, we have:

2x^2 = 34/3

x^2 = 17/3

Alternate Solution:

Let’s raise each side of x^2 + y^2 = 3 to the second power:

(x^2 + y^2)^2 = 3^2

x^4 + 2x^2y^2 + y^4 = 9

Let’s substitute y^4 = x^4 - 25:

x^4 + 2x^2y^2 + x^4 - 25 = 9

2x^4 + 2x^2y^2 = 34

2x^2(x^2 + y^2) = 34

Recall that x^2 + y^2 was given to be 3, thus:

2x^2 * 3 = 34

x^2 = 34/6 = 17/3

Answer: C

_________________

Scott Woodbury-Stewart

Founder and CEO

GMAT Quant Self-Study Course

500+ lessons 3000+ practice problems 800+ HD solutions