Mathematics 10 — Exercise 2.3 · Q8

Full question

If α,β are roots of x² + ax + b = 0 and α²,β² are roots of x² + Ax + B = 0, prove A = 2b − a², B = b².

Step-by-step solution

1. Original sum α+β = −a and product αβ = b.

2. Expand (α+β)² = α²+2αβ+β², so the new sum is α²+β² = a²−2b.

3. The coefficient A is minus this sum: A = −(a²−2b) = 2b−a².

4. The new product is α²β² = (αβ)² = b², so B = b².

Answer

The final answer is shown below.