Using the quadratic formula:
x=(a2+1±√(a4+2a2+1-4a2))/(2a),
x=(a2+1±√(a4-2a2+1))/(2a),
x=(a2+1±(a2-1))/(2a),
x=(a2+1+a2-1)/(2a)=2a2/2a=a or x=(a2+1-a2+1)/(2a)=2/(2a)=1/a.
Roots are a and 1/a.
Without using the formula: solve for x:
ax2-(a2+1)x+a=0,
x2-(a+1/a)x+1=0. The x term is the sum of the roots and 1 is their product.
(a)(1/a)=1 so the roots are a and 1/a.
The quadratic factorises a(x-a)(x-1/a)=a(x2-(a+1/a)x+1)=ax2-(a2+1)x+a.