Mathematical notation and symbols
Introduction
This introductory block is here to remind you of some important notations and conventions used in Mathematics and Statistics.
Numbers and common notations
The numbers are called natural numbers. These are denoted by (whereas the set denotes all natural numbers including the number 0).
Integers are denoted by and include negative numbers too:
Numbers that can be expressed as a ratio of two integers (that is, of the form where and are integers, and ) are said to be rational.
Numbers such as cannot be expressed as a ratio of integers; thus they are called irrational.
The set of real numbers includes both rational and irrational numbers and is denoted by .
The reciprocal of any number is found if we divide 1 by that number. For example, the reciprocal of is and the reciprocal of is 3. Note that the old denominator has become the new numerator, and the old numerator has become the new denominator.
The absolute value of a number can be thought of as its distance from zero. This is denoted by vertical lines around the number. For example, (read "the absolute value of 6") is , and is again.
The factorial of a non-negative integer number is denoted by (read " factorial") and is the product of all positive integers less than or equal to . For example . We also define to be equal to .
Using symbols
Mathematics provides a very rich language for the communication of different concepts and ideas. In order to use this language it is of high importance to appreciate how symbols are used to represent physical quantities, and to understand the rules and conventions that have been developed to manipulate them.
The choice of which letters or symbols to use is up to the user, although it is helpful to choose letters that have some meaning in any particular context. For example, if we wish to choose a symbol to represent the temperature in a room we might choose the capital letter . Usually the lowercase letter is used to represent time. Since both time and temperature can vary we refer to and as variables. In a particular calculation some symbols represent fixed and unchanging quantities and we call these constants.
We often reserve the letters , and to stand for variables and use the earlier letters of the alphabet, such as , and , to represent constants. The Greek letter is used to represent the constant which appears in the formula for the area of the circle. Other Greek letters are frequently used, and for reference the Greek alphabet is given below.
| Letter | Upper case | Lower case | Letter | Upper case | Lower case |
|---|---|---|---|---|---|
| Alpha | Nu | ||||
| Beta | Xi | ||||
| Gamma | Omicron | ||||
| Delta | Pi | ||||
| Epsilon | or | Rho | |||
| Zeta | Sigma | ||||
| Eta | Tau | ||||
| Theta | or | Upsilon | |||
| Iota | Phi | or | |||
| Kappa | Chi | ||||
| Lambda | Psi | ||||
| Mu | Omega |
Mathematics is a very precise language and care must be taken to note the exact position of any symbol in relation to any other. If and are two symbols, then the quantities , and can all mean different things. In the expression , is called a superscript while in the expression it is called a subscript.
If the letters and represent two numbers, then their sum is written as .
Subtracting from yields . This quantity is also called the difference of and .
The instruction to multiply and is written as where usually the multiplication sign is omitted and we simply write . This quantity is called the product of and .
Note that is the same as . Because of this we say that multiplication is commutative.
Multiplication is also associative. When we multiply three quantities together, such as , it doesn't matter whether we evaluate first and then multiply the result by , or evaluate first and then multiply the result by . In other words, .
The quantity (or x/y) means that is divided by . In the expression the top line is called the numerator and the bottom line is called the denominator. Division by leaves any number unchanged (i.e. is simply ) while division by is never allowed.
The equals sign, , is used in several different ways:
- It can be used in equations. The left-hand side and right hand side of an equation are equal only when the variable involved takes specific values known as solutions of the equation. For example, in the equation , the variable is and the left-hand side and right-hand side are equal when has the value . If has any other value the two sides are not equal.
- It can be used in formulae. Physical quantities are often related through a formula. For example, the formula of the length, , of the circumference of a circle expresses the relationship between the circumference of the circle and its radius . It specifically states that . When used in this way the equals sign expresses the fact that the quantity on the left is found by evaluating the expression on the right.
- It can also be used in identities. At first sight an identity looks like an equation, except that is true for all values of the variable. For example, is true for all values of the variable .
The sign is read "is not equal to". For example it is correct to write .
The notation (read "Sigma notation") provides a convenient way of writing long sums. The sum is written using the capital Greek letter sigma, , as .
The notation (read "product notation") provides a convenient way of writing long products. The product is written using the capital Greek letter Pi, , as .
Inequalities
Given any two real numbers and , there are three mutually exclusive possibilities:
( is greater than ),
( is less than ), or
( is equal to ).
The inequality in the first two cases is said to be strict.
The case where " is greater than or equal to " is denoted by . Similarly, we have that .
In these cases, the inequalities are said to be weak.
Some useful relations are:
If and ; then .
If ; then for any .
If ; then for any positive .
If ; then for any negative .
Laws of indices
Indices or powers provide a convenient notation when we need to multiply a number by itself several times. the number is written as and read "5 raised to the power of 3". Similarly we could have
More generally, in the expression , is called the base and is called the index or power.
There are a number of rules that enable us to manipulate expressions involving indices. These rules are known as the laws of indices and they occur so commonly that it is worthwhile to memorise them.
The laws of indices state:
(when multiplying two numbers that have the same base we just add their indices)
(when dividing two numbers that have the same base we subtract their indices)
(if a number is raised to a power and the result itself is raised to a power, the two powers are multiplied together)
Note that in all the previous rules the base was the same throughout.
Two important results that can be derived from these laws are that:
(any number raised to the power of is ), and
(any number raised to the power of is itself).
A generalisation of the third law states:
- (when two numbers, and , are multiplied together and they are raised to the same power, each number is raised to that power and they can then be multiplied together).
Negative indices
A number can be raised to a negative power. This is interpreted as raising the reciprocal number to the positive power. For example, .
Generally, we have that and .
Fractional indices
Let's now consider the expression . Using the third law of indices we can write it as
So is a number that when it is raised to the power of equals . That means that it could be or . In other words is a square root of , that is . There are always two square roots of a non-zero number, and we write .
Similarly, we have that
so that is a number that when it is raised to the power of equals . Thus is the cubic root of , that is which is equal to . Each number has only one cubic root.
Generally, we have that is the -th root of , that is defined as .The generalisation of the third law of indices states that . By taking and we have that .
Polynomial expressions
An important group of mathematical expressions that use indices are known as polynomial expressions. Examples of polynomials areNotice that they are all constructed using non-negative whole-number powers of the variable. Recall that and so the number appearing in the first expression can be thought of .
A polynomial expression takes the formwhere are all constants called the coefficients of the polynomial. The number is also called the constant term. The highest power in a polynomial is called the degree of the polynomial. Polynomials with degree , , and are known as cubic, quadratic, linear and constant respectively.
Tasks
Task 1
Write out explicitly what is meant by the following:
(a)
(b)
(c)
(d)
(e)
(f)
Show answer
Task 2
By writing out the terms explicitly show that
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Task 3
Write out fully, the following expressions:
(a)
(b)
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Task 4
Simplify the following expressions:
(a)
(b)
Show answer
(a)
(b)
Task 5
Remove the parentheses from the following expressions:
(a)
(b)
(c)
Show answer
(a)
(b)
(c)
Task 6
Show that is equal to .
Show answer
Task 7
Write each of the following expressions using a positive index:
(a)
(b)
(c)
Show answer
Task 8
Simplify the following expressions:
(a)
(b)
Show answer
(a)
(b)
Task 9
Evaluate the following:
(a)
(b)
Show answer
(a)
(b)
Sets
Introduction to sets
- A set is a well-defined, unordered collection of objects. We typically use curly brackets to denote sets, for example .
- The objects that make up the set are also known as elements of the set.
- If is an element of , we can say that belongs to and write (the symbol reads "belongs to" or "in"). If, on the other hand, an element does not belong to we can write . To give an example, for , , but .
- A set may contain finitely many or infinitely many elements.
- A set with no elements is called the empty set and is denoted by the symbol .
- The number of elements within a set is called the cardinality of the set and is denoted by or .
- Given sets and , we say that is a subset of if every element of is also an element of . We can then write . In that case, we can also say that is a superset of ; and write it as . The diagram below (which is known as a Venn diagram) illustrates the definition.
Subset
- Given sets and , their union is the set of elements that are either in or (or in both).
- Given sets and , their intersection is the set of elements that are both in and .
- A set is called the complement of if it contains all the elements that do not belong to it. The complement of is written as (or or ).
- Given two sets, and , the difference contains all elements of that are not contained in . The set difference can be, more formally defined as the intersection of and the complement of , .
Example 1
Let's assume that I asked people if they like dogs or cats. of them said they like dogs, of them told me they like cats while there were people who like both dogs and cats.
If we denote as and the sets referring to the people who like cats and dogs respectively; then we are given the following information:, and . Also, there are are 43 people in total. This information is shown on the diagram below (which is known as a Venn diagram).
We will first use the Venn diagram to work out the number of people who like dogs or cats (or both), i.e. we will find the number of elements (cardinality) of the union .
31 people like dogs, cats or both. Note that when we calculated the number people who like dogs or cats, we had to subtract the number of people who like both dogs and cats. The reason for this is that when we add the number of people who like dogs to the number of people who like cats, we have counted those who like both dogs and cats twice. Hence we have to subtract their number, so that we count everyone only once.
Next we will use the Venn diagram to find the number of people who do not like dogs, i.e. the cardinality of the complement .
20 people do not like dogs.
Example 2
Let's assume that we want to look at the set of some European capital cities that start with the letter L. In this case we have
Now, let's assume we are interested to look at the set of the European capital cities that have a small population (let's say less than people). In this case we have
The corresponding Venn diagram is shown below.
Let's now look at some rule for manipulating expressions involving sets.
- For any two sets, and , the intersection of and is the same as the intersection of and , . Similarly, the union of and is the same as the union of and , . This property (which also holds for addition and multiplication of real numbers) is called commutativity. Note that the set difference is not commutative: is not the same as (you can illustrate this on a Venn diagram).
- For any three sets, , and ,i.e. the order in which we take intersections does not matter. This property is called associativity in Mathematics. We can use Venn diagrams to illustrate why this property holds. The left column below identifies , whereas the right column identifies . We can see both are the same.
- Similarly, for any three sets, , and ,i.e. the order in which we take unions does not matter either. This property is called associativity in Mathematics.
- Furthermore, for any three sets, , and , This property is called distributivity in Mathematics. We can again try to understand this rule by identifying both the left-hand sided and the right-hand side in a Venn diagram.
- Similarly, for any three sets, , and ,
You might at first be slightly puzzled by these rules, but you have already been familiar with most of them. You know them from doing arithmetic with numbers. Just think of unions as additions and intersections as multiplications.
| Name | Rule for sets | Corresponding rule for add'n and mult'n of numbers | Example |
|---|---|---|---|
| Commutativty | |||
| Associativity | |||
| Distributivity | No equivalent rule | Addition is not distributive over multiplication. | |
The final two rules are important rules, but have no equivalent in terms of addition or multiplication. They are called De Morgan's laws.
- The complement of the union of two sets equals the intersection of their complements, i.e.
- De Morgan's laws also state that the complement of the intersection of two sets equals the union of their complements, i.e.
Again, you have already been familiar with these rules. This time not from arithmetic with numbers, but from logical statements involving the English words "not" (complement), "or" (union) as well as "and" (intersection).
| De Morgan Law | Logical equivalent | Example |
|---|---|---|
not (S or T) = | "Alice does not like shellfish or tuna." is equivalent to | |
(not S) and (not T) | "Alice does not like shellfish and she does not like tuna." | |
not (S and T) = | "Bob is not both short and tall." is equivalent to | |
(not S) or (not T) | "Bob is either not short or he is not tall." |
Actually, all the rules we have seen so far, not just De Morgan's laws (including commutativity, associativity and distributivity) hold for logical statements involving "and" and "or".
So far we have defined all sets by listing their entries.We can also define sets by stating a property that lets us determine what is an element of the set. For example, the set S consisting of all number which are at least 1 can be written as .
Example 3
If we want the set to be comprised of all the numbers that are greater than , we have Similarly, if we want a set which consists of all the numbers that are smaller than , we have Finally, consists of all the numbers between and .
As the sets , , and are intervals it is easiest to illustrate them on the real line.
All intervals are infinite sets: they contain an infinite number of elements. The reason for this is the infinite resolution of the real numbers. Between any two real numbers lie an infinite number of other real numbers.
We will come back to intervals at the end of this section.
Example 4
Sets containing integers
Assume that set .
We now look at two sets and , which are subsets of : consists of all multiples of whereas consists of all multiples of .
Let's first try to write down the elements of . They are
Note that is a finite set of integers, and not an interval.
The elements of are the elements of which are multiples of 3:
The elements of are the elements of which are multiples of 2:
The elements of the union are those numbers which are multiples of 2 or 3:
The elements of the intersection are those numbers which are multiples of 2 and of 3:
Bounded sets
A set of real numbers is bounded above if there exists a real number that is greaterthan or equal to every element of the set. That is, for some we have for all .The number , if it exists, is called the upper bound of the set .
A set of real numbers is bounded below if there exists a real number that is less than orequal to every element of the set. That is, for all . The number , if it exists, iscalled the lower bound of the set .
A set that is bounded below and bounded above is called a bounded set.
Example 5
(a) The set of all real numbers is neither bounded below, nor bounded above. For every real number we can think of, there is another real number (for example ), which is larger than it. Similarly, for every real number we can think of there is another real number (for example ), which is smaller than it.(b) The set of all natural numbers is bounded below, as all natural numbers are greater or equal to . Just like the real numbers, is not bounded above.(c) Just like the real numbers, the set of all integers , is neither bounded below, nor bounded above.(d) The set , however, is both bounded above and bounded below. is a lower bound for the set, and is a upper bound for the set, as(e) The set is bounded below, but not bounded above.
Maximum and minimum
If a set :
has a largest element , we call the maximum element of the set.
has a smallest element , we call the minimum element of the set.
Example 6
(a) The set of all real numbers , neither has a minimum, nor does it have a maximum.(b) The set of all natural numbers has the minimum , as no natural numbers are less than .(c) The set of all integers has neither minimum or maximum.(d) The set is a very interesting case. We have just seen that it has a lower bound () and an upper bound (). Whilst is also its maximum, it has no minimum, despite having a lower bound. As increases, is getting smaller and smaller, but it will never be exactly .(e) The set has no minimum either, despite having a lower bound (). The lower bound is not an element of the set. Note that this would be different if we looked at the set . In that case, the lower bound is an element of the set (as we have replaced the sign by a sign), and hence 2 is the minimum of the set.
Can you see a relationship between the existence of bounds and of a minimum or maximum?
The general rules are:
Finite sets have a maximum if and only if they have an upper bound. Similarly, they have a minimum if and only if they have a lower bound.
Infinite sets are more complicated. If an infinite set has a minimum (maximum) then it also has a lower bound (upper bound), but the converse is not true. We have seen infinite sets that were bounded below but that did not have a minimum. Similarly, there can be infinite sets that are bounded above but do not have a maximum.
Intervals
If and are two real numbers, the set of all numbers that lie in between and is called an interval.An open interval does not contain its boundary points. The following is an open interval
A closed interval contains its boundary points. The following is a closed interval
Finally, we may have a half-open interval like
The intervals listed previously are all bounded. The following intervals are all unbounded.
The figure below illustrates the definition of these intervals.
Self help
Below is a list of resources you can access for additional reading.
Tasks
Task 11
Let the set and . Which of the following is true?
(a) is a subset of (b) is a subset of (c) is a subset of and is a subset of (d) Neither is a subset of nor is a subset of
Show answer
The correct answer is (d). Neither is a subset of nor is a subset of .
Task 12
For and , find .
Show answer
Functions
Introduction to functions
- A function is a rule that associates a unique value with any element of a set. A function, , from a set to a set defines a rule that assigns for each a unique element .
- It can also be thought of as a rule that operates on an input , sometimes called the argument of the function, and produces an output .
- In order for a rule to be a function it must produce a single output for any given input. It is however possible that two (or more) inputs are mapped to the same output.
- The set of all values that it "maps" from is called the domain.
- The set of values it maps to is called the range. The mapping can be denoted as where and are the domain and the range of the function respectively.
Example 7
Suppose we want to have an output that is times the input; we can define the function . In that case, we might also say that the input can only take positive values. That refers to the domain of the function. To find the range of the function, we have to find what are all the possible outputs. Since we are always multiplying a positive number with the number that means that we will always end up with a positive number. That is the range of the function. To check if this a function or not we can clearly see that for any specific value of we will always get the same output (e.g. if then is always equal to ). The value of the output is often called the value of the function.
Example 8
Let's consider the rule that maps temperature measurements from the Celsius scale to the Fahrenheit scalewhere is the Celsius measurement and the associated value in Fahrenheit.
So, if the temperature in Glasgow is Celsius degrees and we want to find the equivalent temperature in Fahrenheit; we just need to find . Replacing with the value in the previous function will give us Fahrenheit degrees.
In literature, it is common to use instead of . It is just a different notation, nothing else changes.
Graphs are a convenient and widely used way of portraying functions. By inspecting a graph it is easy to describe a number of properties of the function being considered. For example, where is the function positive? and where is is negative? Is it increasing or decreasing?
In order to plot the graph you can start using different values for the argument and then write down the values of the function . Afterwards, you can draw a pair of axes and connect all the previous pairs.
Standard classes of functions
Algebraic functions: are functions that can be expressed as the solution of a polynomial equation with integer coefficients. Some examples of polynomial functions are:
- Constant function ,
- Identity function ,
- Linear function ,
- Quadratic function ,
- Cubic function .
Transcendental functions are functions that are not algebraic. Some examples are:
- Exponential function ,
- Logarithmic function ,
- Trigonometric functions like .
Example 9
Below are the graphs of six functions.
We will now try to match them to the following six functions:
- The function at the top-left (Panel A) is a parabola. Two functions of the functions listed above are also parabolae, so we need to find out which one is which. We can see from the plot that f(0)=0, hence this must be .
- The function to its right (Panel B) is constant, and the only constant function is .
- The function at the top-right (Panel C) is a linear functionwith a positive slope (i.e. the function increases as is increasing), i.e. it must be .
- The function at the bottom-left (Panel D) is another parabola, this time with . Hence, it must be .
- The function to its right (Panel E) is an oscillating function, so it must be the trigonometric function .
- The function at the bottom-right (Panel F) is a non-linear odd function, i.e. it satisfies . Hence it must be .
Inverse of a function
We have seen that a function can be regarded as taking an input , and processing it in some way to produce a single output . A natural question is whether we can find a function that will reverse the process. If we can find such a function it is called an inverse function to and is given the symbol . Do not confuse the with an index or a power. Here, the superscript is used purely as the notation for the inverse function.
To find the inverse of a function you can:
Replace with (this will make the rest of the process easier).
Replace every with a and replace every with an .
Solve the equation from Step 2 for , and finally
replace with .
Can you find the inverse of the function where is the Celsius measurement and the associated value in Fahrenheit?
(Hint: You will have to end up with a function where the input is the Fahrenheit measurement and is the associated value in Celsius.)
A function , defined on domain , is one-to-one if never has the same value for two distinct points in . This means that if we choose two different values and such that ; then we will have . Functions such as do not possess an inverse since there are two values of associated with each (e.g. if we use and we have that ). Note that every one-to-one function has an inverse.
Tasks
Task 13
Consider the function defined as , . Find .
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Task 14
If (for ), show that .
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We could find the inverse by solving for , but it is easier to show that , i.e. applying the function "undoes" the effect of applying the function .
Similarly,
(Note that , i.e. we know that cannot be negative.)
Task 15
If (for ), then for what value is ? (i.e. solve for )
Show answer
VideoVideo model answers for partDuration1:17
If
then this is (assuming that ) equivalent to
We can now bring all terms involving to the left-hand side and keep all terms not involving on the right-hand side, giving
We can now pull out the common factor of on the left-hand side, giving
Dividing both sides by gives
Task 16
Find the inverse of the following functions,
(a) (b) \ where \ (c)
Self help
Weblink 4
Introduction to functions
Exponents and logarithms
Introduction to exponents
An exponent is another name for a power (or an index). Expressions involving exponents are called exponential expressions. For example, consider any positive real number . We define the exponential function to base as .
Sometimes the irrational number is used as the base for the exponential function (called as the natural exponential function). In that case we have . The domain of the function is all the real numbers (this can be also written as while the range of the function is only the positive real numbers .
The figure below shows the graph of the natural exponential function for .
Since the exponential function is monotonic it has an inverse. Its inverse function is the natural logarithm, but more on that later.
Some properties of the exponential function can be seen from the figure:
As becomes large and positive, increases without bound. We express this mathematically as as (the symbol reads "goes to" and the symbol refers to infinity).
As becomes large and negative, approaches . We write as .
The function is never negative.
The property that increases as increases is referred to as exponential growth.
Laws of exponents
The laws of indices and the rules of algebra apply to exponential expressions.
Introduction to logarithms
Logarithms are an alternative way of writing expressions that involve powers, or indices.
Consider the expression . Remember that is the base and is the power. Another way to write this expression is and is stated as "log to base of is ". We see that the logarithm, , is the same as the power in the original expression. The base in the original expression is the same as the base of the logarithm.
The two statements are equivalent. If we write one of them, we are automatically implying the other.
In general, if is a positive constant and then .The number is called the base of the logarithm. In practice most logarithms are to the base or . The latter logarithms, to base , are called natural logarithms and are usually denoted by or (without a subscript, though some use to refer to the logarithm with base , which we will denote by ).
We define the logarithmic function to base as . If we use as the base we then have the natural logarithmic function that we write as . We can see the results of plotting the graphs of the logarithmic (with base ) and the natural logarithmic function for in the figure below.
Some properties of the logarithmic function can be seen from the figure:
As increases, both and increase indefinitely. We write this mathematically as as , as .
As approaches both and approach minus infinity. We express this as as , as .
The value for both functions is when the argument takes the value .
Both functions are not defined when is negative or zero. The domain of the function is all the positive real numbers, , while the range of the function is all real numbers .
Laws of logarithms
Just as expressions involving indices can be simplified using appropriate laws, so expressions involving logarithms can be simplified using the laws of logarithms. These laws hold true for any base. However it is essential that the same base is used throughout an expression before the laws can be applied.
Tasks
Task 17
Write the following using logarithms:
(a) (b) (c) (d) (e)
Show answer
VideoVideo model answers for partDuration0:55
(a) (b) (c) (d) (e)
Task 18
Write the following using indices:
(a) (b) (c)
Show answer
VideoVideo model answers forDuration0:50
(a) (b) (c)
Task 20
Evaluate
Show answer
Task 21
If - = 0, then what value does take?
Show answer
Using that , the statement is equivalent to
Exponentiating both sides gives
yielding
Self help
Quadratic Equations
Quadratic Functions
We will take a closer look at quadratics, which are polynomials of degree 2. You may come across quadratics in different forms, such as:
- an expression
- a function
- an equation
- etc.
where , and are constants and
Solving Quadratic Equations
These can be solved by factorising, using the quadratic formula or by completing the square. We will focus on the first two methods only.
Solving quadratic equations by factorising:
To solve a quadratic equation , let's first start by factorising . To do this we need to find two numbers, and , that multiply to give us (i.e., factors of ) and add to give us so that we end up with:
It's easier to illustrate this using an example.
Example 10
Suppose we wish to solve the equation
We first start by factorising . Two numbers that multiply to give and add to give , are and .
So we have:
The original equation can be rewritten in terms of the product of the factors:
We now have two factors, and multiplying together to give us . If we multiply two (or more) factors and get a zero result, then we know that at least one of the factors is itself equal to zero. So we have:
and are called the roots of the quadratic equation .
Visually, finding the roots of a quadratic corresponds to finding the intersections of the graph of the quadratic function with the x-axis.
Let's try a few more examples.
Example 11
Solve .
In this example we note that the second sign is negative, which means that the signs in the brackets must be different (unlike the previous example where the signs in both brackets were positive).
The pairs of numbers multiplying to give are or .
Since the numbers add to give , we go with and .
So factorises to give
This means that:
Example 12
Solve .
We note that the numbers in the brackets should multiply to give , which means both numbers should be positive or both should be negative.
The pairs of numbers multiplying to give are or .
Since the numbers add to give , we choose and .
So factorises to give
This means that:
Example 13
Solve
Here we note that the numbers in the brackets should multiply to give .The pairs of numbers multiplying to give are ; ; or .
Since the numbers add to give , we choose and .So factorises to give
This means that:
Example 14
Solve The pairs of numbers multiplying to give are ; and . We go with since they multiply to give and add to give .
So factorises to give
This means that:
Here we have an example of repeated roots. Visually, this corresponds to the graph of the quadratic function touching (but not intersecting) the x-axis at the repeated root.
Using the Quadratic Formula
Many quadratic equations cannot be solved by factorisation easily, sometimes because they do not have simple factors. The way round this is to use the quadratic formula.The solution of an equation is given by:
The symbol means that the square root has a positive and a negative value, both of which must be used in solving for x.
Let's try out an example that cannot be easily factorised.
Example 15
Solve .
We substitute , and into the quadratic formula and get:
Here we have or to d.p.
Task 23
Have a go at some of these questions, using the quadratic formula if needed and rounding to d.p. if appropriate:
(a)
(b)
(c)
(d)
(e)
Show answer
The answers to the first two parts are:
(a) Using the quadratic formula with , and
so we have and .
We could have also factorised (as and ), immediately revealing the roots.
(b) Using the quadratic formula with , and
so we have a repeated root at .
We could have also factorised (as and ), immediately revealing the repeated root.
The answers to the other parts are:
(c) or .(d) or .(e) or .
For a quadratic equation , is called the discriminant.
- If , the roots are real and distinct;
- If , the roots are real and repeated;
- If , the roots are complex;
This last option takes us into the realm of Complex Numbers. For example, consider the quadratic
This means that .
At this point, one would say that as we cannot find the square root of a negative number, this problem cannot be solved. However, a way to overcome this issue is that we define an imaginary number as
and this allows us to now find . Bearing in mind that (or that , we can say that
Numbers formed by combining the imaginary number with real numbers are called complex numbers. These numbers take the form where is the real part and is the imaginary part; and are real numbers and is the imaginary number.
Example 16
Consider the quadratic .
If we draw the function we can see that its minimum lies above the x-axis, i.e. it never intersects the x-axis.
Using the quadratic formula, we get:
Notice that the roots have the same real part and different signs on the imaginary part. We call these conjugate pairs.
