Solution of a cubic equation (Part 1)

In elementary algebra it is of prime interest to find the solution of a cubic equation.In this post and few other posts we will find the solution of a cubic equation. As from the previous post we know that the cubic is symmetric with respect to some x = a. In this post let us find the condition when the cubic expression has the middle point on line y=0 and what are the solutions of the corresponding cubic equation.

Let the cubic expression be y = ax³ + bx² + cx+ d.
Differentiating y w.r.t. x we get y' = 3ax² + 2bx + c. Here the sum of the slopes is −2b/3a. Now when we take a particular slope and put it in the equation of y' then we get a quadratic equation. We know that a quadratic equation has two roots. This implies that as the middle point of the curve has both the slopes equal as that point has a unique value. So slope is half of the value −2b/3a i.e. −b/3a. Which we have seen in the previous post. Now we have to find the value of x for which the slope is −b/3a. At this point we know that the difference between the slopes is zero as it has unique value so the rate of change of slope is zero. Differentiating again w.r.t. x we get y'' = 6ax + 2b. We find that the point where rate of change of slope is zero is -b/3a. Equating 6ax + 2b to 0. x has value −b/3a at the point where the rate of change of slope is zero.

Now we will find the value of y for this point. It is
y₁ = a(−b/3a)³ + b(−b/3a)² + c(−b/3a) + d
y₁ = −b³/(27a²) + b³/9a² − bc/3a + d
y₁ = 2b³/27a² − bc/3a + d
When this value is zero then the roots can be found very easily. In this case the center of the curve lies on the axis y = 0. I will tell what are the properties of the roots at the last.
when y₁ = 0 then d = bc/3a − 2b³/27a²

Now let us find the value of x where the slopes are zero. We get
3ax² + 2bx + c = 0
x = [−2b ± √(4b² − 12ac)]/6a
x = [−b ± √(b² − 3ac)]/3a
x₁ = [−b − √(b² − 3ac)]/3a
x₂ = [−b + √(b² − 3ac)]/3a
The center root lies at the middle of the points where the slopes are zero.
So,if δ is the distance of the middle root from the points where the slopes are zero then x₁ + δ = x₂ − δ.
δ = (x₂ − x₁)/2
δ = [√(b² − 3ac)]/3a
Hence the middle root is β = −b/3a.
Let the other roots be at a distance p from the middle root. Then
(β − p)(β + p)β = −d/a
(β − p)(β + p)β =− [bc/3a − 2b³/27a²]/a
(β − p)(β + p) = − [bc/3a − 2b³/27a²]/aβ
(β² − p²) = [bc/3a − 2b³/27a²]3/b
(β² − p²) = [c/a − 2b²/9a²]
(β² − p²) = [9ac − 2b²]/9a²
p² = β² + [2b²−9ac]/9a²
p² = b²/9a² + [2b² − 9ac]/9a²
p² = [3b² − 9ac]/9a²
p = ±√[3b² − 9ac]/3a
Hence the roots are
α = { − b − √[3b² − 9ac]}/3a;
β = −b/3a;
γ = { − b + √[3b² − 9ac]}/3a

Properties of the roots

The outer roots are equidistant form the middle root.

The roots are
α = { −b − √[3b² − 9ac]}/3a;
β = −b/3a;
γ = { − b + √[3b² − 9ac]}/3a
if the equation ax³ + bx² + cx + d = 0
satisfies d = bc/3a − 2b³/27a²

Solution of a cubic equation. A special solution and Why -b/3a?

While solving a cubic equation we come across Tschirnhaus transformation. This transformation is used to express a cubic equation in depressed form. In Tschirnhaus transformation we substitute x by t−b/3a. But why it is −b/3a that is the unique value for it. In this post I will show you why such transformation is useful in case of a depressed cubic.

Look at the following figure.

By looking at the figure we may think the the curve is symmetrical about some x = a. But let us prove it. The expression of the cubic is f(x) = ax³ + bx² + cx + d. Differentiating f(x) with respect to x we get f '(x) = 3ax² + 2bx + c. This is the rate of change of curve. Differentiating again we get f ' '(x) = 6ax + 2b. This is rate of change of slope of the curve. This value is zero for x = −b/3a. f '(x) is quadratic in x so it has two roots or in other words we can express it as the product of two linear expressions. The sum of these two roots is −2b/3a. Let the two roots of the expression f '(x) be α and β. Then α + β = −2b/3a. As we can see that the sum of slopes is twice of the rate of change of slope. So the curve is symmetrical about the line x = −b/3a. On both sides of this line at equal distance the value of slope has the same modulus.

The curve given in the figure is y = x³ − 6x² + 5x − 4. Here (−b/3a) = 6/3 = 2. Shifting the origin to X = x − 2 we get
y = (x+2)³ − 6(x+2)² + 5(x+2) − 4
y = x³ + 8 + 6x² + 12x − 6x² − 24x − 24 + 5x + 10 v 4
y = x³ − 7x − 10

So we see that when we shift the origin to −b/3a we get a cubic whose graph is symmetrical about the axis x=0.

Probability and Statistics

Probability and Statistics is one of the main branch of mathematics. It deals with population and chance. In statistics we read about stats of different kinds their arrangement, collection, classification, analysis and interpretation. In probability we deal with the study of chances and their ways. Probability mainly consists of permutations and combinations. There are many uses of probability in statistics. Let us have a look at what we study in probability and statistics.

Statistics has two different meanings. One is the data and other is a discipline. The data is facts and figures which we come across different conditions in our life and discipline refers to the collection, analysis and interpretation of those data. In the beginning of the study of statistics we deal with its different types of distributions. Then we deal with measures of central tendency. They are mean, median and mode. After it we deal with measures of dispersion. They are range, mean deviation and standard deviation. This study gives us a broad view of the data. The mean is the value which is the most probable when the whole data is taken into account. Median tells us about the middle, quartile, decile and percentile. Mode tells us which value occurs the most. The measure of dispersion tells us about the trend of the data, how much it is away with respect to the mean and the square of the mean. Then we deal with skewness and kurtosis. A measure of skewness would tell us how far the frequency curve of the given frequency distribution deviates from a symmetric one. On the other hand, a measure of kurtosis gives us some information about the degree of flatness (or peakedness) of the frequency curve. Then we deal with how one type of data is related to other type of data. Such study is dealt in correlation and regression. These things deals with statistics. To study more in statistics we need knowledge of probability.

In probability we deal with Random Experiment, sample space, event and algebra of events. Then we deal with probability of discrete sample space. After it we study about probability distributions. These are Bernoulli, Binomial, Multinomial, Hypergeometric, Geometric, Poisson and Negative Binomial. These are standard probability distributions. Then we deal with univariate distributions. After it we deal with standard continuous distributions. They are Normal, Gamma or Exponential, Beta Distributions. Then we deal with bivariate distributions. Then comes functions of random variables chi-square distribution, t-distribution and F-distribution. Then we end the topic by studying the methods used in factories.

Complex numbers and its origin

When we hear of complex numbers, one thing comes to our mind and it is something complex. One thing I would like to tell that complex numbers are not as complex as it seems to be. Even Euler wanted it to be taught in schools. It is a combination of symbols, idea and presentation. the evolution of complex numbers is attributed to the equation x² + 1 = 0. As we can see from the above equation that the square of a number is negative. And we know that the square of any integer cannot be negative. To solve such type of equations we use imaginary numbers. The basic of imaginary numbers is √-1. We represent it by i. One of the most interesting thing in the complex numbers is the product of two complex number i. i×i = √-1 × √-1 is not 1 as can be thought. The reasoning of the above is √-1 × √-1 = √(-1×-1) = √1 = 1. But we must keep in mind that now the number can not be thought as per the rules of real algebra. Any negative number n under the root sign can not be thought as a combination of a number, sign and a root. But it is to be interpreted as a sign i√n. Where the number n denotes the magnitude of the negative number. So the new reasoning is i×i = i² = -1.

A complete imaginary number consists of real as well as imaginary parts.But one thing which comes to our mind is what is the representation of these numbers. To show such kind of numbers we use Argand plane or complex plane. A complex number has two parts the real part and the imaginary part. The real part is denoted on the real axis and imaginary part is denoted on the imaginary axis. The real part is equivalent to the x-axis and the imaginary part is equivalent to the y-axis of the Cartesian Coordinates.

Euler established a relationship between e, i, cos θ and sin θ. He found that they form a definition and it is
e^iθ = cos θ + i sin θMuch of the mathematics of complex numbers is result of identity. We find that e^iθ = cos θ + i sin θhas no simple proof but as we expand e^iθ we find that its representation is like this
e^iθ = 1 + iθ + i²θ² + i³θ³ + i⁴θ⁴ + ...
= 1 + iθ − θ² − iθ³ + θ⁴ + ...
= (1 − θ² + θ⁴ ...) + i(θ − θ³ + ...)
We know that (1 − θ² + θ⁴ ...) is expansion of cos θ and (θ − θ³ + ...) is the expansion of sin θ
= cos θ + i sin θ

Similarly by induction we can prove that (cos nθ + i sin nθ) = (cos θ + i sin θ)ⁿ.

There are many such results which form the basis of complex numbers. But the origin of complex numbers if the result of deep knowledge of mathematics as it takes ideas from different fields.

Solution of Quadratic by graph

The solution of a quadratic equation, cubic equation is possible with graph. I have found a method by which we can find the solution of a quadratic, cubic, etc, equations with the help of graph when one of the solution is known. The solution consists of finding a equation which relates the roots of the equation. It is very easy to find such a solution. Although the equation consists of only two roots but when we examine the roots then we can find all the roots of the equation. We will find the expression for quadratic equation in this post and discuss cubic equation in the next post. The method goes like this.

1. Let the equation be ax² + bx + c = 0
2. Take two roots of the equation. Let it be x₁ and x₂.
3. Form one equation with one root.
   ax₁² + bx₁ + c = 0 ---(i)
   Form one more equation with other root.
   ax₂² + bx₂ + c = 0 ---(ii)
   
4. Subtract (ii) from (i) and solve it
   (ax₁² − ax₂²) + (bx₁ − bx₂) + c − c = 0
   a(x₁² − x₂²) + b(x₁ − x₂) = 0
   (x₁ − x₂)[a(x₁ + x₂) + b] = 0
   Suppose both the roots are different, then
   [a(x₁ + x₂) + b] = 0
   (x₁ + x₂) = −b/a
   
You will get a relation in the roots.
5. Plot the relation on a graph by taking x₁ to be y and x₂ to be x.
   The relation becomes y = −x − (b/a)

Graphic Solution of two quadratic expressions y = x² + 5x + 6 and y = x² − 5x + 6 and their respective relation equation between roots are given. The roots can be found by checking where the quadratic graph cuts the x-axis.

The two straight lines gives the solution of the equations x² + 5x + c₁ = 0 and
x² − 5x + c₂ = 0. If we know one root of the equation then we can find the other root by looking at the point's corresponding coordinate point where one coordinate is equal. For example as one roots of x² + 5x + 6 = 0 is 2 so the corresponding point on the curve taking x = 2 is (2,3) so the other root is 3. We can also find the roots by looking at the point (a,b) where the area formed by the rectangle between x-axis, y-axis, x = a and y = b have area equal to the constant term. The condition for the roots for a particular equation is that the roots has product equal to c/a, where c is the constant term and a is coefficient of x².

Solution of a Cubic by graph

As we have seen in the post 'Solution of Quadratic by Graph' we can represent solution of a quadratic equation. Similarly we can represent solution of a cubic equation also by graph. The method is the same we represent a relation between two roots. The relation gives us a graph which has solution as the value of x coordinate and the corresponding y coordinate. In the case of cubic we get an ellipse. Let us first find the relation between two roots of a cubic equation.

1. Let the equation be ax³ + bx² + cx + d = 0
2. Take two roots of the equation. Let it be x₁ and x₂.
3. Form one equation with one root.
   ax₁³ + bx₁² + cx₁ + d = 0 ---(i)
   Form one more equation with other root.
   ax₂³ + bx₂² + cx₂ + d = 0 ---(i)
   
4. Subtract (ii) from (i) and solve it
   (ax₁³ − ax₂³) + (bx₁² − bx₂²) + (cx₁ − cx₂) + d − d = 0
   a(x₁³ − x₂³) + b(x₁² − x₂²) + c(x₁ − x₂) = 0
   (x₁ − x₂)[a(x₁² + x₂² + x₁x₂) + b(x₁ + x₂) + c] = 0
   Suppose both the roots are different, then
   [ax₁² + ax₂² + ax₁x₂ + bx₁ + bx₂ + c] = 0
   [ax₁² + ax₁x₂ + bx₁ + ax₂² + bx₂ + c] = 0
   [ax₁² + x₁(ax₂ + b) + (ax₂² + bx₂ + c)] = 0
   solving in x₁ we get,
   x₁ = (1/2a)( − (ax₂ + b)±√((ax₂ + b)² − 4a(ax₂² + bx₂ + c)))
   x₁ = (1/2a)( − (ax₂ + b)±√(−3a²x₂² − 2abx₂ + b² − 4ac))
   
You will get a relation in the roots.
5. Plot the relation on a graph by taking x₁ to be y and x₂ to be x.
   The relation becomes y = (1/2a)( − (ax + b)±√( − 3a²x² − 2abx + b² − 4ac))
   Divide the relation into two parts one with + sign and other with − sign. Plot them.

Solution of the cubic equation in the above graph is 1,2 and 3. As we can see from the graph that the solution exist where the line x=1 or x=2 or x=3 cuts the curve. When we look at the line x = 1, then we find that the line cuts the curve at y=2 and y=3. Similarly we see that the curve is cut by the line x=2 at y=1 and y=3. So is the case with x=3, where y=1 and y=2. So the solutions of the equation is the points satisfying the condition that the product of the roots is equal to the negative of the constant term divided by coefficient of x³.

Similarly on the guidelines for the previous graph we find that the solution is 1,2 and 4.

When the graph is plotted then the roots of the family of curves ax³ + bx² + cx + k = 0, where k is parameter is given by the curve which is like the ellipse. If one root is known to be α. Then other two roots can be found by looking at the points where the line x = α intersect the ellipse. By noting down the two values of y where the line x = α intersect we get the three roots. The two values of y and one value of x or two values of x and one value of y are the three roots of the curve.

Elementary Algebra

Elementary Algebra is based on unknown. Although number of value to be found or the number of solutions is known in advance by looking at the equations. These solutions are masked by the variables. A variable can take any value and the values which satisfy the equations are known as the solutions. A linear equation in one unknown has one solution. Similarly a linear equation in two variables has one solution if two equations are given,a linear equation in three unknowns also has one solution and is solvable if three equations are given in three unknowns. The set of equations which are used are called a system of equations. But a quadratic equation has two solutions and a cubic equation has three solutions. A variable raised to n power has n solutions.

Let us talk about what is taught in elementary algebra. We will talk as a whole that is what exist in elementary algebra from schools to colleges. The first thing to be taught is about sets. Set is the language of mathematics. This thing occurs in every branch of mathematics. The next thing is solution of linear equations, quadratic equations, cubic equations and quartic equations. The cubic equations are solved by Cardano's Method. The quartic equations are solved by Ferrari's or Descartes method. The next thing which comes under elementary algebra is complex numbers. The rise of complex numbers is attributed to the equation in which the square of a variable is equal to a negative number. Euler gave a definition by which he joined the complex number with trigonometry. It is worth to note it here.
e^iθ = cos θ + i sin θ
One consequence of this formula is e^iπ = −1
There is one more formula given by De Moivre and it is
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
This formula is used in the expansion of cos θ and sin θ and it also ha many other uses.

The next thing we are taught is the system of equations. As described in the first paragraph it has solutions if a set of equations are given. The solution is possible under certain conditions and has a unique solution under a particular condition. The set of equations can have a unique solution or a dependent solution or no solution. The solution of linear systems can be found by Cramer's rule or by use of matrices. There is also matter related to inequalities. The inequalities known to the ancients (Inequality of the means, Triangle inequality), Less ancient inequalities (Cauchy-Schwarz inequality, Weierstrass’ inequalities, Tchebyshev’s inequalities).

Elementary algebra is taught from the beginning because it has vast applications in other fields. The expressions which we come across in elementary algebra takes shape of functions in calculus. The solution of quadratic, cubic and quartic equations are used to solve the maximum and minimum problems. The factor theorem taught also comes handy in calculus as well as statistics. Binomial theorem, multinomial theorem are used in statistics.

Every student will find it interesting if he finds instances where it is used in our life. I will try my best to bring those things come to life on this blog.