Tìm x biết: (x - 2)(x - 4)(x - 5)(x - 10) - 54x^2 = 0
Câu hỏi:
Tìm x biết: (x – 2)(x – 4)(x – 5)(x – 10) – 54x2 = 0.
Trả lời:
(x – 2)(x – 4)(x – 5)(x – 10) – 54x2 = 0
⇔ [(x– 2)(x – 10)][(x – 4)(x – 5)] – 54x2 = 0
⇔ (x2 – 12x + 20)(x2 – 9x + 20) – 54x2 = 0
Đặt x2 – 12x + 20 = t
Khi đó ta có:
t(t + 3x) – 54x2 = 0
⇔ t2 + 3xt – 54x2 = 0
⇔ t(t – 6x) + 9x(t – 6x) = 0
⇔ (t + 9x)(t – 6x) = 0
⇔ (x2 – 18x + 20)(x2 – 3x + 20) = 0
⇔ \(\left[ \begin{array}{l}{x^2} - 18x + 20 = 0\\{x^2} - 3x + 20 = 0\end{array} \right.\)
Nếu x2 – 18x + 20 = 0
⇔ (x – 9)2 – 61 = 0
⇔ (x – 9)2 = 61
⇔ \(\left[ \begin{array}{l}x = 9 + \sqrt {61} \\x = 9 - \sqrt {61} \end{array} \right.\)
Nếu x2 – 3x + 20 = 0
⇔ \[{\left( {x - \frac{3}{2}} \right)^2} + \frac{{71}}{4} \ge \frac{{71}}{4} > 0\] nên phương trình vô nghiệm.
Vậy x = \(\left\{ {9 + \sqrt {61} ;9 - \sqrt {61} } \right\}\).
