# Tag Archives: Algebra

# Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers with a bizarre property that they can be formed by adding up all the smaller numbers that make up their divisors. The number $6$ is the first perfect number, because it may be divided by […]

# Method of Exhaustion

Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using the $\textit{Method on Exhaustion}$: at this stage, you achieve a more accurate figure. To find an increasingly accurate solution to $\sqrt{2}$, for example, he produced a ladder of numbers. Starting with $1$ and $1$ as the first row, he then added those […]

# The Golden Ratio

Since the days of old, artists, as well as mathematicians, have known that there is a special, aesthetically pleasing rectangle with width $1$, length $x$, and the following property: When a square of side $1$ is removed, the rectangle that remains has the same proportions as the original rectangle. Since the new rectangle has a […]

# Index Notation

A convenient way to write a product of $\textit{identical factors}$ is to use $\textbf{exponential}$ or $\textbf{index notation}$.Rather than writing $5 \times 5 \times 5 \times 5$, we can write this product as $5^4$. The small $4$ is called the $\textbf{exponent}$ or $\textbf{index}$, and the $5$ is called the $\textbf{base}$.If $n$ is a positive integer, then […]

# The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$, $-$, $\times$, $\div$, $($ $)$, etc by the Sign of One. Click the numbers below to see the answers. The first one is done for you.

# Surd Equations Reducible to Quadratics

Surd Equations Reducible to Quadratic for Math Algebra is done squaring both sides for removing surds and radical expressions. Make sure to check whether the solutions are correct by substituting them into the original surd equations. Question 1 Solve \( x = \sqrt{x+2} \). \( \begin{aligned} \displaystyle \require{color}x^2 &= x+2 &\color{red} \text{square both sides} \\x^2 […]

# Trigonometric Equations Reducible to Quadratics

Trigonometric Equations Reducible to Quadratic for Math Skills are based on trigonometric identities such as;$$\sin^2{x} + \cos^2{x} = 1 \1 + \cot^2{x} = \csc^2{x} \\tan^2{x} + 1 = \sec^2{x} \$$ Question 1 Solve \( 2 \cos^2{x} – 3 \cos{x} + 1 = 0 \) for \( 0^\circ \le x \le 360^\circ \). \( \begin{aligned} \displaystyle(\cos{x}-1)(2 […]

# 4 Important Types of Absolute Value Equations

There are 4 main types of absolute value equations regarding whether there are; absolute value and a static value absolute value and an expression involving unknown pronumerals two absolute values in both sides two absolute values and a value Type 1: One Absolute Value and a Constant Solve \( | x-2 | = 5 \). […]

# Inequalities using Arithmetic Mean Geometric Mean

Arithmetic Mean of \(a\) and \(b\) is always greater than or equal to the Geometric Mean of \(a\) and \(b\), for all positive real numbers with with equality if and only if \(a = b\). This is also called AM-GM (Arithmetic Mean Geometric Mean) inequality. \(\require{color}\) $$ \begin{aligned} \frac{a + b}{2} \ge \sqrt{ab} \text{ or […]