Question

If \(x^2 + y^2 = 25\) and \(xy = 12\), then what is the value of \(x^4 + y^4\)?

• A. \(289\)
• B. \(193\)
• C. \(337\)
• D. \(241\)

Hint:

Use algebraic identities like \( x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \) when both \(x^2 + y^2\) and \(xy\) are known.

Answer 1

The Correct Option is C

Step 1: Use identity to relate \(x^4 + y^4\) with \(x^2 + y^2\) and \(xy\). We start with the identity: \[ x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2 \] Step 2: Use the given: \[ x^2 + y^2 = 25, \quad xy = 12 \Rightarrow x^2y^2 = (xy)^2 = 144 \] Step 3: Substitute in the identity: \[ x^4 + y^4 = (25)^2 - 2(144) = 625 - 288 = 337 \].