Let α, β be the roots of the equation x² - 4λ x + 5 = 0 and α, γ be the roots of the equation x² - ( 3√2 + 2√3 )x + 7 + 3λ √3 = 0, λ > 0. If β + γ = 3√2, then (α + 2β + γ )² is equal to __________.

Solution

Maths sir · 4 steps

Add the two sums of roots: β + γ is given, so you get α in terms of λ straight away.

1. Write the sums. From the first equation, α + β = 4λ and αβ = 5. From the second, α + γ = 3√2 + 2√3 and αγ = 7 + 3λ√3.
2. Find alpha. Adding the two sums gives 2α + (β + γ) = 4λ + 3√2 + 2√3. Put β + γ = 3√2 to get α = 2λ + √3, and so β = 4λ - α = 2λ - √3.

   αβ = (2λ + √3)(2λ - √3) = 4λ² - 3 = 5 ⇒ λ² = 2

3. Fix lambda and check. Since λ > 0, λ = √2. Then α = 2√2 + √3 and γ = √2 + √3, and αγ = 4 + 3√6 + 3 = 7 + 3√6, which matches 7 + 3λ√3.
4. So. α + 2β + γ = (α + β) + (β + γ) = 4√2 + 3√2 = 7√2. Its square is 49 × 2 = 98.

The idea to keep. When two quadratics share a root, write both sums and products first. Adding or subtracting the sums often isolates the common root before you touch the equations themselves.

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