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olve the logarithmic equations $\log_3(x+1)=\log_4(x+8)$ The bases of this equation is different so Let $\log_3(x+1)=y$...............................(i) And Let $\log_4…

Solve the indicial equation $4^\frac{1}{x}-6^\frac{1}{x}=9^\frac{1}{x}$ To solve the equation, simply divide through by any of the term since the indices are all given i…

Solving quadratic equations using the Poh-Shen Log method. The Poh-Shen Loh method is another method for solving quadratic equations, the method was pioneered by Prof. P…

An object is thrown vertically upwards with an initial velocity, $u$ of $20m/s$. The motion of the object follows the differential equation $\frac{ds}{dt}=u-gt$, where $…

Below is a beautiful proposition on matrix eigenvalues. It says find the minimal eigenvalue of the matrix $A^{2021}$. We find the eigenvalue using the determinant for $|…

The question below was obtained from one of the past questions for the Skoltech institute entrance examination for admission into computational science masters program. …

Prove of the Cassini-Samson Identity The Cassini-Samson Identity is an identity that involves the Fibonnaci numbers, it states that $F_{n-1}F_{n+1}-F_{n}^2=(-1)^n$ for $…

Determine the value of the following series $\sum^{\infty}_{n=1}(\frac{4}{n^2+4n+3}-9^{-n+2}4^{n+1})$ Solution To determine their value, we must check if the series is a…

The question says evaluate the polynomial $\frac{x^2}{ax+b}$ using long division. Therefore, $f(x)=x^2=(ax+b)(\frac{x}{a}-\frac{b}{a^2})+\frac{b^2}{a^2}$.

Lucas number: another beautiful patterns of sequence Formula for obtaining the Lucas numbers The Lucas numbers are numbers that form Fibonacci integer sequence, each Luc…

The harmonic functions and construction of analytic functions.  In this video series Dr. Kamran Khan takes us through harmonic functions and construction of analytic fun…

How to curry and uncurry a function $\mathbb{R}\rightarrow\mathbb{R}^N$.  The functions $A\rightarrow{B}^C$ corresponds to functions of the form $A\times{C}\rightarrow{B…

How do you proof $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$? It is not always true. To see the clear picture, we solve some clear examples. Remember, $\sqrt{\frac{a}…