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Completing the Square (Step by Step) - ChiliMath

Details: How to Complete the Square In a regular algebra class, completing the square is a very useful tool or method to convert the quadratic equation of the form also known as the “standard form”, into the form which is known as the vertex form. Find the Vertex Form of using Completing the Square Example 1: … Completing the Square (Step by Step) Read More »

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Simplifying Radical Expressions - ChiliMath

Details: Simplifying Radical Expressions A radical expression is composed of three parts: a radical symbol, a radicand, and an index In this tutorial, the primary focus is on simplifying radical expressions with an index of 2. This type of radical is commonly known as the square root. Components of a Radical Expression Starting with a single radical expression, … Simplifying Radical Expressions Read

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Solving Multi-Step Equations - ChiliMath

Details: Solving Multi-Step Equations The word “multi” means more than two, or many. That’s why solving multi-step equations are more involved than one-step and two-step equations because they require more steps. The main goal in solving multi-step equations, just like in one-step and two-step equations, is to isolate the unknown variable on one side of the … Solving Multi-Step Equations Read

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Solving Absolute Value Equations - ChiliMath

Details: Solving Absolute Value Equations. Solving absolute value equations is as easy as working with regular linear equations. The only additional key step that you need to remember is to separate the original absolute value equation into two parts: positive and negative (±) components.Below is the general approach on how to break them down into two equations:

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Solving Two-Step Equations - ChiliMath

Details: The USUAL way of solving a two-step equation: Note: This is the “usual” method because most of the two-step equations are solved this way.Notice that Step 2 can alternatively be replaced by Step 3 which are the same essentially. 1) First, add or subtract both sides of the linear equation by the same number. 2) Secondly, multiply or divide both sides of the linear equation by the same number.

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Creating Math Website Using WordPress - ChiliMath

Details: Math Website Using WordPress That Generates Passive Income Post Updated: May 5, 2020 Disclosure: Some links included in this post are affiliate links which mean that I will get some monetary compensation whenever you click the links and make a purchase. I do NOT write any paid reviews. All the products and services, free or paid, … Creating Math Website Using WordPress Read More »

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Solving Quadratic Equations by Square Root Method - ChiliMath

Details: Key Strategy in Solving Quadratic Equations using the Square Root Method. The general approach is to collect all {x^2} terms on one side of the equation while keeping the constants to the opposite side. After doing so, the next obvious step is to take the square roots of both sides to solve for the value of x.Always attach the \pm symbol when you get the square root of the constant.

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Combining or Condensing Logarithms - ChiliMath

Details: Combining or Condensing Logarithms The reverse process of expanding logarithms is called combining or condensing logarithmic expressions into a single quantity. Other textbooks refer to this as simplifying logarithms. But, they all mean the same. The idea is that you are given a bunch of log expressions as sums and/or differences, and your task is … Combining or Condensing Logarithms Read

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Properties of Real Numbers - ChiliMath

Details: The Properties of Real Numbers In this lesson, we are going to go over the different properties of real numbers (ℜ). Understanding the properties of real numbers will help us simplify numerical and algebraic expressions, solve equations, and more as you progress in studying algebra. For clarity, “properties” in this context refer to the characteristics … Properties of Real Numbers Read

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There are infinitely many prime numbers. - ChiliMath

Details: Prove: There are infinitely many prime numbers. How do we even start proving that something is infinite? For instance, how do we show that there are infinitely many prime numbers? The premise in itself is so mind-boggling because the concept of infinity is vast and at its very core, very daunting to comprehend. However, we … There are infinitely many prime numbers. Read More »

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Multiplying Rational Expressions - ChiliMath

Details: Multiplying Rational Expressions Rational expressions are multiplied the same way as you would multiply regular fractions. Nothing more, nothing less. As you may have learned already, we multiply simple fractions using the steps below. Review the Steps in Multiplying Fractions Multiply the numerators. Multiply the denominators Simplify the “new” fraction by canceling common factors. Most

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Plotting Points on a Graph or XY-plane - ChiliMath

Details: Plotting Points on a Graph In this tutorial, I have prepared eight (8) worked out examples on how to plot a point in a Cartesian plane (named in honor of French mathematician Renè Descartes). To plot a point, we need to have two things: a point and a coordinate plane. Let’s briefly talk about each one. A Point … Plotting Points on a Graph or XY-plane Read More »

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Literal Equations - ChiliMath

Details: Solving Literal Equations Literal equations, simply put, are equations containing two or more variables. Your goal is to solve for just one variable with respect to others. If you know how to solve regular equations, then I guarantee you that solving literal equations will be a breeze. What is a Literal Equation? A literal equation is an … Literal Equations Read More »

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Partial Fraction Decomposition - ChiliMath

Details: Partial Fraction Decomposition This method is used to decompose a given rational expression into simpler fractions. In other words, if I am given a single complicated fraction, my goal is to break it down into a series of “smaller” components or parts. Previously on adding/subtracting rational expressions, we want to combine two or more rational expressions into a … Partial Fraction

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Absolute Value Inequalities - ChiliMath

Details: Solving Absolute Value Inequalities In this lesson, we are going to learn how to solve absolute value inequalities using the standard approach usually taught in an algebra class. That is, learn the rules and apply them correctly. There are four cases involved when solving absolute value inequalities. CAUTION: In all cases, the assumption is that the … Absolute Value Inequalities Read More »

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If n^2 is even, then n is even. - ChiliMath

Details: Prove: Suppose is an integer. If is even, then is even. Let be an integer. We want to show that whenever is even, then must be even. Just a heads-up, the result of this theorem is significant because it will help you prove a more important fact [in the future] that the square root of … If n^2 is even, then n is even. Read More »

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Mathematical Induction for Divisibility - ChiliMath

Details: Mathematical Induction for Divisibility. In this lesson, we are going to prove divisibility statements using mathematical induction. If this is your first time doing a proof by mathematical induction, I suggest that you review my other lesson which deals with summation statements.The reason is students who are new to the topic usually start with problems involving summations followed by

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Basic or Simple Proportions - ChiliMath

Details: Basic or Simple Proportions Suppose there are two ratios and . They can be written as fractions and , respectively. Now, if we set these two ratios equal to each other then it becomes a proportion. Ways to Write a Proportion Proportion is a statement showing that two ratios are equal. There are two ways … Basic or Simple Proportions Read More »

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Coin Word Problems - ChiliMath

Details: Coin Word Problems. The secret to success in solving coin word problems is to be able to set up the correct systems of equations and precisely solve it using the substitution method or at times, the elimination method.. It’s also worth pointing out that the presentation of the algebraic expressions in coin word problems are a bit different and not so straightforward compared to what we’re

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Simplifying Complex Fractions - ChiliMath

Details: Simplifying Complex Fractions When a “normal” fraction contains fractions in either the numerator or denominator or both, then we consider it to be a complex fraction. This type of fraction is also known as a compound fraction. There are two methods used to simplify such kind of fraction. After going over a few examples, you should … Simplifying Complex Fractions Read More »

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Solving Exponential Equations using Logarithms - ChiliMath

Details: How to Solve Exponential Equations using Logarithms In our previous lesson, you learned how to solve exponential equations without logarithms. This time around, we want to solve exponential equations requiring the use of logarithms. Why? The reason is that we can’t manipulate the exponential equation to have the same or common base on both sides … Solving Exponential Equations using

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Lessons - ChiliMath

Details: FREE MATH LESSONS For almost seven (7) years, ChiliMath has strictly focused on algebra and to date, has over 150 algebra lessons available at no cost to students. The algebra lessons or skills are divided into three (3) categories: Introductory Algebra Intermediate Algebra Advanced Algebra Complete List of Algebra Lessons And at last, after a few … Lessons Read More »

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Factorial Notation and Formula - ChiliMath

Details: Factorial Notation, Formula, and Basic Examples When I first encountered an algebra problem with exclamation mark “!“, I thought it was a trick question. I didn’t know how to handle it because I had no idea what it meant. As you know, symbols in math are everything. The key is to recognize that each mathematical symbol … Factorial Notation and Formula Read More »

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List of Odd Numbers - ChiliMath

Details: List of Odd Numbers Feel free to review the concept of an odd number. Click the image below to take you to my lesson about odd numbers. If you’re looking for a comprehensive list of odd numbers from 1 to 1,000, this is the place for you! I listed the odd numbers into ten (10) … List of Odd Numbers Read More »

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Generating Pythagorean Triples - ChiliMath

Details: Generating Pythagorean Triples using a Formula You can generate a Pythagorean Triple using a formula. The proof of why the following formula works all the time is beyond the scope of this lesson. For our purposes, let us call this the “Pythagorean Triple Formula”. Just a bit of caution, this formula can generate either a … Generating Pythagorean Triples Read More »

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Horizontal Line Test - ChiliMath

Details: Horizontal Line Test. The horizontal line test is a convenient method that can determine whether a given function has an inverse, but more importantly to find out if the inverse is also a function.. Remember that it is very possible that a function may have an inverse but at the same time, the inverse is not a function because it doesn’t pass the vertical line test.

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Arithmetic Sequence Formula - ChiliMath

Details: The Arithmetic Sequence Formula If you wish to find any term (also known as the term) in the arithmetic sequence, the arithmetic sequence formula should help you to do so. The critical step is to be able to identify or extract known values from the problem that will eventually be substituted into the formula itself. … Arithmetic Sequence Formula Read More »

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Composition of Function - ChiliMath

Details: Composition of Function In this lesson, I will go over eight (8) worked examples to illustrate the process involved in function composition. If we are given two functions, it is possible to create or generate a “new” function by composing one into the other. The step involved is similar when a function is being evaluated for … Composition of Function Read More »

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Descartes' Rule of Signs - ChiliMath

Details: Descartes’ Rule of Signs The purpose of the Descartes’ Rule of Signs is to provide an insight on how many real roots a polynomial may have. We are interested in two kinds of real roots, namely positive and negative real roots. The rule is actually simple. Here is the Descartes’ Rule of Signs in a nutshell. … Descartes’ Rule of Signs Read More »

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Solving Linear Inequalities - ChiliMath

Details: Solving Linear Inequalities Most of the rules or techniques involved in solving multi-step equations should easily translate to solving inequalities. The only big difference is how the inequality symbol switches direction when a negative number is multiplied or divided to both sides of an equation. In this lesson, I will go over seven (7) worked … Solving Linear Inequalities Read More »

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Cramer's Rule with Three Variables - ChiliMath

Details: Cramer’s Rule for a 3×3 System (with Three Variables) In our previous lesson, we studied how to use Cramer’s Rule with two variables.Our goal here is to expand the application of Cramer’s Rule to three variables usually in terms of \large{x}, \large{y}, and \large{z}.I will go over five (5) worked examples to help you get familiar with this concept.

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Converse, Inverse, and Contrapositive of Conditional

Details: Converse, Inverse, and Contrapositive of a Conditional Statement What we want to achieve in this lesson is to be familiar with the fundamental rules on how to convert or rewrite a conditional statement into its converse, inverse, and contrapositive. But first, we need to review what a conditional statement is because it is the foundation … Converse, Inverse, and Contrapositive of Conditional

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Multiplying Polynomials - ChiliMath

Details: Multiplying Polynomials To be successful in multiplying polynomials, you will need to apply the knowledge learned from the following two prerequisite topics: Product Law of Exponent The rule states that when multiplying two exponential expressions with the same base, simply copy the common base then add their exponents. Adding and Subtracting Polynomials The key idea is that we … Multiplying

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If n^2 is odd, then n is odd. - ChiliMath

Details: Prove: Let be an integer. If is odd, then is odd. We assume that is an integer. Our goal is to show that is odd is a sufficient condition to justify that is odd. Down the road, you will appreciate the result of this theorem as it will be very useful in proving the fact … If n^2 is odd, then n is odd. Read More »

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Solving Exponential Equations without Logarithms - ChiliMath

Details: Key Steps in Solving Exponential Equations without Logarithms. Make the base on both sides of the equation the SAME. so that if \large{b^{\color{blue}M}} = {b^{\color{red}N}}. then {\color{blue}M} = {\color{red}N}. In other words, if you can express the exponential equations to have the same base on both sides, then it is okay to set their powers or exponents equal to each other.

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Proofs of Logarithm Properties - ChiliMath

Details: Proofs of Logarithm Properties or Rules The logarithm properties or rules are derived using the laws of exponents. That’s the reason why we are going to use the exponent rules to prove the logarithm properties below. Most of the time, we are just told to remember or memorize these logarithmic properties because they are useful. … Proofs of Logarithm Properties Read More »

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Ratios - ChiliMath

Details: Ratios In general, a ratio is an expression that shows the relationship between two values. It tells us how much of one thing is there as compared to another. There are two “kinds” of ratios: “part to part” and “part to whole“. KINDS OR TYPES OF RATIOS Part-to-Part Ratio In a nutshell, the ratio of some value to some value is … Ratios Read More »

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Verifying if Two Functions are Inverses of Each Other

Details: Verifying if Two Functions are Inverses of Each Other. Previously, you learned how to find the inverse of a function.This time, you will be given two functions and will be asked to prove or verify if they are inverses of each other. But how? The procedure is really simple. But before I do so, I want you to get some basic understanding of how the “verifying” process works.

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Solving Quadratic Equations by Factoring Method - ChiliMath

Details: How to Solve Quadratic Equations using Factoring Method This is the easiest method of solving a quadratic equation as long as the binomial or trinomial is easily factorable. Otherwise, we will need other methods such as completing the square or using the quadratic formula. The following diagram illustrates the main approach to solving a quadratic … Solving Quadratic Equations by Factoring

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Simplifying Factorials with Variables - ChiliMath

Details: Simplifying Factorials with Variables In this lesson, we will learn how to simplify factorial expressions with variables found in the numerator and denominator. We want to generate common factors in both locations so that they can be canceled. That’s ultimately our goal. The key is to compare the factorials and determine which one is larger … Simplifying Factorials with Variables Read More »

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How to Graph a Line - ChiliMath

Details: Three Ways How to Graph a Line. Graphing a line in the xy-axis is one of the most basic skills in algebra that you should learn. In this lesson, we are going to go over three methods that are commonly used. Learn each method and you’ll be able to decide which one works better depending on the given problem.

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Scalar Multiplication - ChiliMath

Details: Scalar Multiplication: Product of a Scalar and a Matrix. There are two types or categories where matrix multiplication usually falls under. The first one is called Scalar Multiplication, also known as the “Easy Type“; where you simply multiply a number into each and every entry of a given matrix.. The second one is called Matrix Multiplication which is discussed on a separate lesson.

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Solving Quadratic Equations by Completing the Square

Details: How to Solve Quadratic Equations using the Completing the Square Method If you are already familiar with the steps involved in completing the square, you may skip the introductory discussion and review the seven (7) worked examples right away. The key step in this method is to find the constant “” that will allow us to express … Solving Quadratic Equations by Completing the Square Read

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Proof: √(2) is irrational. - ChiliMath

Details: Prove: The Square Root of , , is Irrational. Proving that is irrational is a popular example used in many textbooks to highlight the concept of proof by contradiction (also known as indirect proof). This proof technique is simple yet elegant and powerful. Basic steps involved in the proof by contradiction: Assume the negation of … Proof: √(2) is irrational. Read More »

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Inverse of a 2x2 Matrix - ChiliMath

Details: Inverse of a 2×2 Matrix. In this lesson, we are only going to deal with 2×2 square matrices.I have prepared five (5) worked examples to illustrate the procedure on how to solve or find the inverse matrix using the Formula Method.. Just to provide you with the general idea, two matrices are inverses of each other if their product is the identity matrix.

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Relations and Functions - ChiliMath

Details: Relations and Functions Let’s start by saying that a relation is simply a set or collection of ordered pairs. Nothing really special about it. An ordered pair, commonly known as a point, has two components which are the x and y coordinates. This is an example of an ordered pair. Main Ideas and Ways How … Relations and Functions Read More »

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Solving Quadratic Equations by the Quadratic Formula

Details: How to Solve Quadratic Equations using the Quadratic Formula. There are times when we are stuck solving a quadratic equation of the form a{x^2} + bx + c = 0 because the trinomial on the left side can’t be factored out easily. It doesn’t mean that the quadratic equation has no solution.

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Determinant of 3x3 Matrix - ChiliMath

Details: The Formula of the Determinant of 3×3 Matrix. The standard formula to find the determinant of a 3×3 matrix is a break down of smaller 2×2 determinant problems which are very easy to handle. If you need a refresher, check out my other lesson on how to find the determinant of a 2×2.Suppose we are given a square matrix A where,

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Inverse Variation - ChiliMath

Details: Inverse Variation (also known as Inverse Proportion) The concept of inverse variation is summarized by the equation below. Key Ideas of Inverse Variation We say that varies inversely with if is expressed as the product of some constant number and the reciprocal of . However, the value of can’t equal zero, i.e. . Isolating on one side, it … Inverse Variation Read More »

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Adding and Subtracting Matrices - ChiliMath

Details: Adding and Subtracting Matrices In this lesson, I have prepared seven (7) worked examples to illustrate the basic approach on how to easily add or subtract matrices. If you know how to add and subtract real numbers, this topic should really be a breeze. The only thing required in order to “legally” perform the operations … Adding and Subtracting Matrices Read More »

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