Let x1, x2, x3, x4 be the solution of the equation 4x^4 + 8x^3 - 17x^2 - 12x + 9 = 0

Question

Let x₁, x₂, x₃, x₄ be the solution of the equation 

\(4 \mathrm{x}^{4}+8 \mathrm{x}^{3}-17 \mathrm{x}^{2}-12 \mathrm{x}+9=0\) and 

\(\left(4+x_{1}^{2}\right)\left(4+x_{2}^{2}\right)\left(4+x_{3}^{2}\right)\left(4+x_{4}^{2}\right)=\frac{125}{16} \mathrm{~m}.\)Then the value of m is ________.

Answer 1

Correct answer is: 221

\(4 x^{4}+8 x^{3}-17 x^{2}-12 x+9\)

\(=4\left(\mathrm{x}-\mathrm{x}_{1}\right)\left(\mathrm{x}-\mathrm{x}_{2}\right)\left(\mathrm{x}-\mathrm{x}_{3}\right)\left(\mathrm{x}-\mathrm{x}_{4}\right)\)

Put \(\mathrm{x}=2 \mathrm{i} \) & \( -2 \mathrm{i}\)

\( 64-64 i+68-24 i+9=\left(2 i-x_{1}\right)\left(2 i-x_{2}\right)\left(2 i-x_{3}\right)\)

\(\left(2 \mathrm{i}-\mathrm{x}_{4}\right)\)

\(=141-88 \mathrm{i}\)                  .......(1)

\(64+64 i+68+24 i+9=4\left(-2 i-x_{1}\right)\left(-2 i-x_{2}\right)(-2 i\)\(\left.-\mathrm{x}_{3}\right)\left(-2 \mathrm{i}-\mathrm{x}_{4}\right)\)

\(=141+88 \mathrm{i}\)                 ........(2)

\(\frac{125}{16} \mathrm{~m}=\frac{141^{2}+88^{2}}{16}\)

\( \mathrm{m}=221\)