Let α, β be the roots of the equation x²+2 √2 x-1=0. The quadratic equation, whose roots are α⁴+β⁴ and 1/10(α⁶+β⁶), is:

Solution

Maths sir · 4 steps

Do not hunt for the roots: get α²+β² once, then build the 4th and 6th powers from it.

1. Sum and product. From x²+2√2x-1=0, we get α+β=-2√2 and αβ=-1. So α²β²=1, and α²+β²=(α+β)²-2αβ=8+2=10.
2. Fourth powers. Square the sum of squares: α⁴+β⁴=(α²+β²)²-2α²β²=100-2=98.

   α⁶+β⁶=(α²+β²)³-3α²β²(α²+β²)=1000-30=970

3. New roots. The two new roots are 98 and 1/10× 970=97. Their sum is 98+97=195 and their product is 98× 97=9506.
4. So. The equation is x²-195x+9506=0, which is option (b).

The idea to keep. For any power sum of roots, find α²+β² from the sum and product first. Higher even powers then follow from a²+b² and a³+b³ identities.

Answer from the NTA final answer key. Solution by PrepWiser.