Category Archives: Algebra

Problem1: Let $x_{1}$ and $x_{2}$ be the root of $x^2-7x-9=0$. Find the value of $|x_{1}-x_{2}|$. Solution $$|x_{1}-x_{2}|=\dfrac{\sqrt{b^2-4ac}}{|a|}=\dfrac{\sqrt{(-7)^2-4\cdot(-9)}}{|1|}=\sqrt{85}$$ Problem 2: Find all solutions of $\sqrt{x+10}-\dfrac{6}{\sqrt{x+10}}=5$. Solution Let $y=\sqrt{x+10}$, we have $$y-\dfrac{6}{y}=5$$ Multiplying $y$ on both side of the equation, we have … Continue reading →

Let $P(x)$ is a polynomial with integer coefficients so that $P(d)=\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Find $P(x)$.🔑 Claim: There is no $P(x)$ to satisfy $P(d)=\dfrac{2025}{d}$, where $d$ is a positive divisor of $2025$. Lemma: If $P(x)$ … Continue reading →

Let $n$ be the real number. Define $[n]$ be the integer part of $n$, and $\{n\}$ be the decimal part of $n$. Solve the following equations: $$\begin{array}{ccccccc} \{x\} & + & [y] & + & \{z\} & = 2.9\\ \{y\} … Continue reading →

Find the real solutions for the following equations: $$a^2+b^2\ \ \ \ \ \ \ \ \ \ \ \ \ =1\tag{1}$$ $$b^2+c^2+\sqrt{3}bc=1\tag{2}$$ $$c^2+a^2+\ \ \ \ \ ca=1\tag{3}$$ Click here for the solution. Solution: If $c=0$, based on equation … Continue reading →