Question

Q1. If $x= (2+ \sqrt{5} )^{ \frac{1}{2} } +( 2- \sqrt{5} )^{ \frac{1}{2} }$ and $y=(2+ \sqrt{5}) ^{ \frac{1}{2} } - (2-\sqrt{5} )^{ \frac{1}{2} }$ then evaluate $x^2+y^2$
Q2.  If (x+a) is a factor of the polynomials $x^2+px+q$ and $x^2+mx+n$ the prove: $a= \frac{n-q}{m-p}$

Answer 1

see the attachment for soln.

Answer 2

x² = 2 + √5  + 2 - √5 + 2 (2² - 5)  
y² = 2 + √5 + 2 - √5  - 2 (4 - 5)

x²+y² = 8

x+a is a factor. So  a²  - p a + q = 0   and a² - m a + n = 0 
Subtract one equation from another.
(m-p) a + (q-n) = 0 
a = (n-q )/ (m-p)