Question

Giải phương trình: \(\left(x-2\right)^4+\left(x+2\right)^4=626\)

Answer 1

Áp dụng bảng tam giác Pascal ta có : 

\(\left(x-2\right)^4=x^4-8x^3+24x^2-32x+16\)

\(\left(x+2\right)^4=x^4+8x^3+24x^2+32x+16\)

\(\Rightarrow\left(x-2\right)^4+\left(x+2\right)^4=2x^4+48x^2+32=626\)

\(\Leftrightarrow2x^4+48x^2-594=0\)

\(\Leftrightarrow2x^4-6x^3+6x^3-18x^2+66x^2-594=0\)

\(\Leftrightarrow2x^3\left(x-3\right)+6x^2\left(x-3\right)+66\left(x-3\right)\left(x+3\right)=0\)

\(\Leftrightarrow\left(2x^3+6x^2+66x+198\right)\left(x-3\right)=0\)

\(\Leftrightarrow\left[2x^2\left(x+3\right)+66\left(x+3\right)\right]\left(x-3\right)=0\)

\(\Leftrightarrow2\left(x+3\right)\left(x^2+33\right)\left(x-3\right)=0\)

\(\Rightarrow x=\pm3\)

Vậy nghiệm \(S=\left\{\pm3\right\}\)