Category Archives: Algebra

Find all real values $x$ such that $4^x+6^x=9^x$.🔑 Solutions: Dividing $6^x$ on both side of the equation, we have: $$\Big{(}\dfrac{2}{3}\Big{)}^x+1=\Big{(}\dfrac{3}{2}\Big{)}^x$$ Let $y=\Big{(}\dfrac{2}{3}\Big{)}^x$, we have $$y+1=\dfrac{1}{y}$$ i.e. $$y^2+y-1=0$$ Solving the above equation, we have $$y=\dfrac{-1\pm\sqrt{5}}{2}$$ Because $x$ is real, and $y=\Big{(}\dfrac{2}{3}\Big{)}^x>0$, … Continue reading →

For number pairs $(r_i, c_i)$, $i=1,2,…,n$, where $n\ge 1$, $r_i\ge 0$ and $c_i\ge 0$, they have the following property:$$\sum_{i=1}^{n}r_i=\sum_{i=1}^{n}c_i=n^2$$ Additionally, there exists a positive value of $k$ so that for every $i$ value,$i=1,2,…,n$, the following three inequalities hold:$$r_i\le \dfrac{n^3}{k}\ \ … Continue reading →

Prove that there is no polynomial $P(x)$ with integer coefficients such that $$P(\sqrt[3]{5}+\sqrt[3]{25})=2\sqrt[3]{5}+3\sqrt[3]{25}$$🔑 Proof: Assume that there is a polynomial $P(x)$ of degree $n$, with integer coefficients, and without trivial root of $0$, such that$$P(\sqrt[3]{5}+\sqrt[3]{25})=2\sqrt[3]{5}+3\sqrt[3]{25}$$Therefore, $P(x)$ can be expressed as … Continue reading →