Question

Giải phương trình
\(3^{x-3+\sqrt[3]{7-3x}}+\left(x^3-9x^2+24x+7\right).3^{x-3}=3^x+1\)

Answer 1

Chia 2 vế cho \(3^{x-3}\)

\(\Leftrightarrow3^{\sqrt[3]{7-3x}}+x^3-9x^2+24x+7=3^3+3^{3-x}\)

\(\Leftrightarrow3^{\sqrt[3]{7-3x}}+x^3-9x^2+27x-27=3^{3-x}+3x-7\)

\(\Leftrightarrow3^{\sqrt[3]{7-3x}}+\left(x-3\right)^3=3^{3-x}+3x-7\)

\(\Leftrightarrow3^{\sqrt[3]{7-3x}}+\left(7-3x\right)=3^{3-x}+\left(3-x\right)^3\) (1)

Xét hàm \(f\left(t\right)=3^t+t^3\Rightarrow f'\left(t\right)=3^t.ln3+3t^2>0;\forall t\)

\(\Rightarrow f\left(t\right)\) đồng biến

Nên (1) tương đương:

\(7-3x=\left(3-x\right)^3\)

\(\Rightarrow x\)