Đáp án:
\(\left[ \begin{array}{l}
x = \frac{{ - 5 + \sqrt {17 + 8\sqrt {22} } }}{2}\\
x = \frac{{ - 5 - \sqrt {17 + 8\sqrt {22} } }}{2}
\end{array} \right.\)
Giải thích các bước giải:
\(\begin{array}{l}
x\left( {x + 1} \right)\left( {x + 4} \right)\left( {x + 5} \right) = 84\\
\to \left( {{x^2} + 5x + 4} \right)\left( {{x^2} + 5x} \right) = 84\left( 1 \right)\\
Đặt:{x^2} + 5x = t\\
\left( 1 \right):\left( {t + 4} \right).t = 84\\
\to {t^2} + 4t - 84 = 0\\
Có: Δ'= 4 + 4.84 = 340\\
\to \left[ \begin{array}{l}
t = - 2 + 2\sqrt {22} \\
t = - 2 - 2\sqrt {22}
\end{array} \right.\\
\to \left[ \begin{array}{l}
{x^2} + 5x = - 2 + 2\sqrt {22} \left( 2 \right)\\
{x^2} + 5x = - 2 - 2\sqrt {22} \left( 3 \right)
\end{array} \right.\\
Xét:\left( 2 \right): Δ= 25 - 4\left( {2 - 2\sqrt {22} } \right) = 17 + 8\sqrt {22} \\
\to \left[ \begin{array}{l}
x = \frac{{ - 5 + \sqrt {17 + 8\sqrt {22} } }}{2}\\
x = \frac{{ - 5 - \sqrt {17 + 8\sqrt {22} } }}{2}
\end{array} \right.\\
Xét:\left( 3 \right): Δ= 25 - 4\left( {2 + 2\sqrt {22} } \right) = 17 - 8\sqrt {22} < 0
\end{array}\)
⇒ Phương trình (3) vô nghiệm
\(KL:\left[ \begin{array}{l}
x = \frac{{ - 5 + \sqrt {17 + 8\sqrt {22} } }}{2}\\
x = \frac{{ - 5 - \sqrt {17 + 8\sqrt {22} } }}{2}
\end{array} \right.\)
