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How to solve equations of 1st, 2nd and 3rd degree

We may face the task of solving different types of equations while studying mathematics, so in this post, we will see how to solve first, second, and third degree equations. Table of content: Possible cases for the solutions Solving first degree equations Solving second degree equations Solving...

Absolute Value and Square Roots

Absolute values often show up in problems involving square roots. That’s because you can’t take the square root of a negative number without introducing imaginary numbers (those involving $i = \sqrt{-1}$ ). Example 1: Simplify $\sqrt{x^{2}}$ This problem looks deceptively simple. Many students...

Absolute Value and Logarithms

Absolute Value and Logarithms Absolute values often turn up unexpectedly in problems involving logarithms. That’s because you can’t take the log of a negative number. Let’s first review the definition of the logarithm function: Logb x = y ⇔ by = x (The double arrow is a bi-conditional, which means...

L’Hospital’s Rule

L’Hospital’s Rule is a useful way to evaluate tricky limits. It is most often used for limits of indeterminate form. The rule is as follows: If $f(x)$ and $g(x)$ are differentiable on some interval around the number $a$ (or if $a=\infty$, $f(x)$ and $g(x)$ are differentiable for all...

Algebraic Groups

Introduction to Algebraic Groups One of the most fundamental algebraic structures in mathematics is the group. A group is a set of elements paired with an operation that satisfies the following four conditions: I. It is closed under an operation (represented here by “+”, although it does not...

Algebraic Rings

Introduction to Algebraic Rings An algebraic ring is one of the most fundamental algebraic structures. It builds off of the idea of algebraic groups by adding a second operation  (For more information please review our article on groups). For rings we often use the notation of addition and...

Modular Arithmetic & Fermat’s Little Theorem

Modular arithmetic is a way of counting in which the numbers wrap around after reaching a certain value. The clock is often used as an analogy. While time always progresses forward, the 12-hour clock “resets” to 1 after passing 12 (13 o’clock is equivalent to 1 o’clock). If we replace 12 with 0,...

Find the Vertex by Using the Quadratic Formula

The Quadratic Formula is primarily used to identify the roots (\(x\)-intercepts) of a quadratic function. What many people don't know is that you can also easily find the vertex of the function by simply looking at the Quadratic Formula! Graphing Quadratic Functions When graphing quadratic...