Basic Algebra
3.7 Quadratic Equations
3.6.1 Examples
The second law of surds is most often used to work out the square roots of fractions.
The first law was often used to split large square roots into smaller ones, by attempting to divide the original number by a perfect square.
1. √
A very important application of this to come later is that of complex numbers.
3.7 Quadratic Equations
A quadratic equation in x is an equation of the form ax2+ bx + c = 0, a 6= 0
where a, b and c are constants (that is, they do not change value as x does.
It’s vital that a is not zero, or the equation collapses into that of a straight line. However, it is quite allowable to have b or c (or both) zero.
3.7.1 Examples
Here are some examples of equations which are, and which are not quadratic equations, shown in table 3.3. Equations 1,2 and 3 are genuine quadratics, even though terms are missing in 2 and 3. Equation 4 is the equation of a straight line, or if you like a = 0 which is not permitted. Equation 5 contains an x3 term, and so is a cubic equation and not a quadratic whose highest term must be x2.
3.7 Quadratic Equations 25
Equation Quadratic? a b c
1 2x2+ 3x − 4 = 0 YES 2 3 -4
2 x2+ 2x = 0 YES 1 2 0
3 x2+ 3 = 0 YES 1 0 3
4 4x + 2 = 0 NO n/a n/a n/a
5 x3+ 3x − 2 = 0 NO n/a n/a n/a
Table 3.3: Examples of quadratic equations
3.7.2 Graphical interpretation
To examine the solutions of this equation, it is helpful to consider the graph of the function given by
y = ax2+ bx + c.
When this graph is plotted it gives a characteristic “U” shaped curve, called parabola. It has many interesting properties, but the most important is that it is symmetrical about a vertical line through the maximum, or minimum point. Our equation corresponds to the above function when y = 0, or to put it another way, the solutions of our equation occur when the curve cuts the x-axis (where y is zero).
Figure 3.1: The quadratic equation
This gives us our first problem, which is that we cannot even be sure that the equation has solutions. If this seems strange or confusing, recall that we never claimed that all equations could be solved. Our problem falls into three categories.
3.7 Quadratic Equations 26
1. The curve cuts the x-axis twice;
2. The curve cuts the x-axis once only (just touching, no more);
3. The curve does not cut the x-axis at all.
This situation is shown in figure 3.1. In this figure all the curves have been drawn as “maximum” parabolas, many quadratics show “minimum”
parabolas but the cases are just the same. The y-axis has been ommitted as it is not relevant to the problem at hand which is determining the number of solutions of the corresponding equation.
3.7.3 Factorization
One approach to try and solve a quadratic is the so called factorization or sum and product method of solution.
We imagine that we can write our equation in the form (x − α)(x − β) = 0.
Why have we written it this way? Well, for a product of two things to be zero, one or other, or both must be zero. Therefore in this case, if the left-hand bracket is zero we see that x = α, and if the right-hand bracket is zero we see that x = β. Therefore α and β are our two solutions.
If we expand these brackets out, we obtain
x2− αx − βx + αβ = x2− (α + β) + αβ = 0
Now we try to compare this to our quadratic equation, but before we do so, we “standardize” our equation, by dividing it by a:
ax2+ bx + c = 0 ⇒ x2+ b ax + c
a = 0.
We now compare the coefficients of the x terms in each equation (assum-ing they are in fact two forms of the same equation). (A coefficient is a number multiplied upon a term).
Comparing we obtain
x2 term 1 = 1 This is why we “standardize”;
x term ba = −(α + β) constant term ca = αβ
3.7 Quadratic Equations 27
The two significant equations are this
−b
a = α + β; c a = αβ
which tell us that the sum of the solutions is −ba and that the product of the solutions is ca. If we can guess two numbers with these properties, we have solved the quadratic equation.
This method has the advantages that the theory is simple and straight forward, but has two crippling disadvantages. Firstly, we have to guess two numbers, and even in simple cases this may be very difficult, in a hard example next to impossible. Secondly, and more importantly, before we start we have no way of knowing how many solutions the equation has. If it has two, this theory works well, if it has one, then α and β have the same value (we say the solution is repeated). However, if there are no solutions, it will be impossible to guess our two numbers, as they do not exist, but this may not be obvious from the start.
3.7.4 Quadratic solution formula
Fortunately, we do have a more flexible method of solution, but its proof is difficult.
The proof is given here, but is not required.
First of all consider the problem, we cannot simply take a square root to remove the square, as the other x term gets in the way. Consequently, we try to absorb the x term and the x2 term into one term. This is a difficult procedure known as completing the square.
Consider
(x + y)2 = (x + y)(x + y) = x2+ 2xy + y2
We wish to let this be our x2 +abx terms. For this to be true, we must let y = 2ab . Note that this does not correspond to exactly what we want:
which is our quadratic, except that the ca term is missing, and the last term above is extra, so we add and subtract these two terms respectively.
x2+ b
3.7 Quadratic Equations 28
which using the laws of surds (see 3.5), yields our final result.
x = −b ±√
b2− 4ac 2a
3.7.5 The discriminant
As well as giving a fool-proof method of solving a quadratic, the solution formula has a small section which tells us some useful information. The section
b2− 4ac
which was the contents of the square root is known as the discriminant of the equation, as it tells us how many solutions the equation has.
b2− 4ac > 0 Two real solutions Case 1 above b2− 4ac = 0 One real solution Case 2 above b2− 4ac < 0 No real solutions Case 3 above
3.7.6 Examples
Here are some examples of quadratic equations and their solutions.
1.
3.7 Quadratic Equations 29
x = −2 ±p22− 4(1)(−3) 2
which when calculated out yields answers of 1 and −3.
2. In this case a = 1, b = −4, c = 4. The solution formula yields x = +4 ±p(−4)2− 4(1)(4)
2
which when calculated out yields answers of 2 and 2. The repeated so-lution is nothing unusual, and indicated that this quadratic has only one solution.
3. In this case a = 1, b = 1, c = 1. We shall follow this case in more detail.
x = −1 ±p12− 4(1)(1) 2
= −1 ±√ 1 − 4 2
= −1 ±√
−3 2
Note that we have a negative square root here. We cannot calculate this, suppose that the answer was a positive number, then when squared we get a positive number, not −3. We also get a positive number when we square a negative number, and we get zero when we square zero. Thus no number can be √
−3.
There are no solutions to this quadratic equation.
3.7.7 Special cases
Observe that if b = 0, or if c = 0 in the quadratic, the equation can be solved more directly. We show how for completeness, although the formula may still be used in these cases.
b = 0
We have here
ax2+ c = 0 ⇒ ax2 = −c ⇒ x2 = −c a Therefore
3.8 Notation 30
x = ± r
−c a.
Note that −ac must be positive, and this will be the case only if a and c have different signs, otherwise there are no solutions. Note also the ± in the formula, solving directly often leads to forgetting the solution coming from the negative branch of the square root.
c = 0
In this case we have
ax2+ bx = 0 ⇒ x(ax + b) = 0 from which we obtain, that either
x = 0 which is one solution, or
ax + b = 0 ⇒ x = −b a
3.8 Notation
We now introduce two more items of notation.
3.8.1 Modulus or absolute value
A very useful idea in mathematics is the notion of absolute value of a real number. If x is a real number, we define the modulus of x, or mod x, denoted
|x| as follows.
|x| = larger of x and −x
Note that this is not the same as the mod operation in computing.
3.8 Notation 31
Examples
Here are some examples
|2| = larger of 2 and −2 = 2;
| − 2| = larger of −2 and 2 = 2;
|0| = larger of 0 and −0 = 0.
In other words, the modulus function simply strips off any leading minus sign on the number.
Alternative definitions
There are some alternative, but equivalent definitions of |x|:
|x| =
x if x ≥ 0
−x if x < 0 and
|x| =√ x2
where we adopt the convention that the square root takes the positive branch only, unless we include ±, which is commonly accepted.
3.8.2 Sigma notation
Suppose that f (k) is some expression involving k (a function of k formally speaking). For example, f (k) could be 2k or 2k + 1 etc. Then, the notation
n
X
k=m
f (k) is a shorthand for the expression
f (m) + f (m + 1) + · · · + f (n)
In other words, we insert the value at the bottom of the sigma into the f expression, then insert that value plus one, plus two, etc., until we reach the value on top of the sigma, and add all the results together.
We will use the symbol ∞ to denote the lack of an endpoint in the sum-mation.
The ranges above and below the sigma are sometimes ommitted when it is clear what is being summed.
This may be easier to understand after some simple examples.