Question

Prove that (2+√x)⁴ + (2-√x)⁴ = 2(16 + 24x + x²)

Answer 1

To prove: (2+√x)⁴ + (2-√x)⁴ = 2(16 + 24x + x²)

Formula used: (i)

ⁿC_r = \(\frac{n!}{(n-r)!(r)!}\)

(ii) (a+b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^n-1b + ⁿC₂a^n-2b² + … +ⁿC_n-1ab^n-1 + ⁿC_nbⁿ 

⇒ 2[(1)a⁴ + (6)a²b² + (1)b⁴] 

⇒ 2[a⁴ + 6a²b² + b⁴] 

Therefore, (a+b)⁴ + (a-b)⁷ = 2[a⁴ + 6a²b² + b⁴]

Now, putting a = 2 and b =(√x) in the above equation. 

(2 + √x)⁴ + (2-√x)⁴ = 2[(2)⁴ + 6(2)² (√x)² + (√x)⁴] 

= 2(16+24x+x²)

Hence proved