-
Algebraic operations are critical to many fields of study. In science, they are used to calculate forces to accelerate a car or determine the rate of a chemical reaction. In engineering, they can be used to design safe and efficient systems. They also underpin algorithms and data structures, which are...
-
Find out how to replace pronumerals with numbers in a formula to get a numerical value for some quantity. In many courses you will be required to use formulae to calculate something of interest. The process of substituting numbers for pro-numerals in an expression or formula is called substitution. Pro-numerals...
-
What do I do with the brackets? Brackets are useful to group numbers, pronumerals and operations together as a whole. Whatever is around the brackets affects all the things inside the brackets. Brackets are commonly used to express mathematical formulae. In order to manipulate expressions containing brackets it is necessary...
-
What is an algebraic fraction? The numerator (top) or denominator (bottom) of a fraction can be in algebraic form involving numbers and variables (represented by pronumerals or letters). We cover the multiplication and division of fractions containing algebraic terms. Simplifying fractions You have probably seen numerical fractions like \(\frac{18}{24}\) before....
-
Let's take a look at algebraic fractions where the denominator is a quadratic expression. Such fractions are common in mathematics and engineering. More complicated algebraic fractions can involve polynomials of any order. However in your studies, it is unlikely that you will have to deal with polynomials higher than quadratics....
-
Find out how to express an algebraic fraction as a sum of simpler algebraic fractions (partial fractions). The method of partial fractions involves breaking up an algebraic fraction into simpler parts that are added together. This is useful in integration and in finding inverse Laplace and Fourier transforms. Adding fractions...
-
Rearranging formulas, also called transposition of formulas, is a necessary skill for most courses. Let's work on some essential skills in manipulating formulas. Introduction Some of the most important equations that we might be required to transpose occur frequently in science, engineering and economics. They are called formulae and give...
-
Learn how to manipulate or rearrange formulas that involve fractions and brackets. Formulas are used in many branches of economics, science and engineering. For example, the formula for simple interest is:1 Note that when we write \(nr\) we mean \(n\times r\). Most of the time we ignore the multiplication sign...
-
Are you still trying to get that variable on its own from the formula, but it is in a tricky place – or maybe it appears more than once? Here we demonstrate manipulating or rearranging complex formulas, with overviews and practice questions. Let's look at some more complicated formulas where...
-
A common factor is a number or pronumeral that is common to terms in an algebraic expression. Removing the common factors allows us to factorise algebraic expressions and write them in a simpler form. Factorisation using common factors is a basic skill in mathematics so we'll discuss that too. Expansion...
-
The idea of a perfect square permeates mathematics and especially algebra. Properties of perfect squares will help you in factorising and expanding algebraic expressions. An understanding of perfect squares is essential for success in mathematics. A perfect square is something like \(5^{2}, x^{2}\) or \(\left(x+1\right)^{2}\). In this module we deal...
-
What is the Difference of two squares (DOTS) rule? If you have a squared expression subtracted from another squared expression, you can factorise this quickly according to the DOTS rule. The difference of two squares formula is commonly used in mathematics. It allows us to factorise terms such as \[\begin{align*}...
-
Quadratic expressions have the general form \[ax^2+bx+c\] where \(a\), \(b\) and \(c\) are real numbers and \(a\neq 0\). Quadratics frequently arise in mathematics, science and engineering. This module explains how to factorise a quadratic into two linear factors. For example \[x^2+5x+6 = \left(x+2\right)\left(x+3\right).\] The general form of a quadratic expression...
-
Quadratic expressions have the general form \[ax^2+bx+c\] where \(a\), \(b\) and \(c\) are real numbers and \(a\neq 0\). Quadratics frequently arise in mathematics, science and engineering. Let's factorise a quadratic into two linear factors using the "completing the square" method. This is a general method that allows any quadratic to...
-
1 An integer is a positive or negative whole number and may include \(0.\) For example in the case that the result was an integer we could have This means that \(6750\div15=450\) or \(6750=15\times450.\) In case the numbers do not exactly divide we could have This means that \[\begin{align*} 6751\div15...
-
What is algebra? Why are there letters in the equation? This page links to resources that will help you answer these questions. Algebraic expressions involve pronumerals (letters) to represent values. Pronumerals can take many different values. We often need to plug our own values into a given formula for calculating...
-
Complex numbers are a group of numbers that help us to get mathematical solutions where real numbers (which includes positive and negative, counting numbers, fractions and decimals) just can’t work. They can be useful in engineering and physical sciences. They have two parts: one part is an imaginary number. (For...
-
Not sure what a linear equation is? Don't know where to start with quadratic or simultaneous equations? The links on this page will take you to resources that will help to fill any gaps in your knowledge. ES1 Linear equations Equations with one variable may be solved using transposition skills...
-
Equations with one variable may be solved using transposition skills to make the variable the subject of the equation. A linear equation has one unknown variable and may be solved by transposing the equation to make the variable the subject of the equation. For example, the equation \[\begin{align*} 3\left(2x-7\right) &...
-
Two equations in two variables are said to be simultaneous if both must be considered at the same time. Two equations in two variables are said to be simultaneous if both must be considered at the same time. An ordered pair \(\left(x,y\right)\) which satisfies both equations is said to be...
-
The general form of a quadratic equation is A X squared plus B X plus C equals zero where A is not equal to one. This tip sheet looks at solutions to quadratic equations using the “null factor law”. General form A equation can be rearranged to the form: \(ax^{2}+bx+c=0\)...
-
The solutions to any quadratic equation can be found by substituting the values a, b, c into the quadratic formula. The solutions to any quadratic equation \(ax^{2}+bx+c=0\) can be found by substituting the values \(a,b,c\) into the quadratic formula: \[ x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}. \] Solutions may be real or complex numbers. In...
-
...
-
Indicial (or exponential) equations have the form ax = b. If we can write b as a number with a base a and an index, then we can equate the indices to find x. If two equal numbers are written to the same base then the indices must be equal....
