If x=1+√2+√3, then x4−4x3−4x2+16x=?
Answer:
8
- Given: x=1+√2+√3
According to the question, we need to find the value of x=1+√2+√3 - x=1+√2+√3⟹x−1=√2+√3
Taking the square on both sides
x2+1−2x=2+3+2×√2×√3 [ Using (a+b)2=a2+b2+2ab]⟹x2−2x−4=2√6 - Again, taking the square on both sides
\begin{align} & \space x^4 + 4x^2 + 16 - 4x^3 + 16x - 8x^2 = 4 \times 6 \space\space\space\space\space [\because (a - b - c)^2 = a^2 + b^2 + c^2 -2ab + 2bc -2ca ] \\ & \implies x^4 - 4x^3 -4x^2 + 16x + 16 = 24 \\ & \implies x^4 - 4x^3 -4x^2 + 16x = 8 \\ \end{align}