Quadratic Formula: What, Why, and How It Changed Mathematics

It’s an essential principle that guides the world of algebra and beyond.
Rupendra Brahambhatt

Almost every student comes across the quadratic formula in mathematics, and it is a popular means to figure out the roots of a quadratic equation. 

In real life, the quadratic formula helps us in determining the area of space, the speed of a moving object, the value of profit gained on a product, and more. Even the path of a space rocket is described in terms of a quadratic equation. Therefore, the quadratic formula not only holds significance in mathematics but also has great utility in the real world.

What is the quadratic formula?

It is often tricky to factorize some specific types of quadratic equations; however, the roots (also called x-intercepts or zeros) of such equations can be easily calculated using the quadratic formula. A quadratic function is graphically represented by a parabola with vertex located at the origin, below the x-axis, or above the x-axis. Therefore, a quadratic function may have one, two, or zero roots.

The quadratic equation is generally given as:

ax2+bx+c = 0

To find the roots of a quadratic function, we can set f (x) = 0, and solve the equation, by completing the square. When we do this, we arrive at the quadratic formula, which is given as:

 x = [-b ± √(b² - 4ac)]/2a

By solving the above equation, the value of x (root) is determined, and the sum of the roots and product of the roots of the equation can also be derived further.

The term b2 −4ac is called the discriminant. The discriminant is important because it tells you how many roots a quadratic function has. Specifically, if:

                                         b2 −4ac < 0 There are no real roots

                                         b2 −4ac = 0 There is one real root

                                         b2 −4ac > 0 There are two real roots

On a graph, for any parabola which is described as y = ax2+bx+cthe roots are the points (or values), where the parabola crosses the x-axis.

  • Discriminant in quadratic formula

The nature of the roots obtained from the quadratic formula is decided by the discriminant (D), which is given as:

D = b2-4ac

When the value of D is zero, the roots are said to be real and equal. If the value of D is positive, the roots obtained are real and unequal, and when D is negative, then roots are complex conjugates, so there are no real roots.

Factorization and completing the square method are two other ways to solve a quadratic equation. However, the quadratic formula is considered more efficient because it is applicable for all the equations and acts as the only single formula that can evaluate the roots in any quadratic equation. Moreover, when compared to the other two methods, it is easier to explain the nature of roots through the quadratic formula, from the value of D.  

  • Types of quadratic equation

A quadratic equation can be written in three different forms:

Standard form: y = ax2 + bx + c 

Factored form: y = (ax + c)(bx + d)

Vertex form: y = a(x + b)2 + c 

You can change a quadratic equation from one form to another depending on your requirement. For example, in case you need to find the zeroes of a standard quadratic equation, you can first change the same into factored form.

Who invented the quadratic formula?

The history of the quadratic formula can be traced all the way back to the ancient Egyptians. The theory is that the Egyptians knew how to calculate the area of different shapes, but not how to calculate the length of the sides of a given shape, e.g. the wall size needed to create a given floor plan.

To solve the practical problem, by around 1500 BC, Egyptian mathematicians had created a table for the area and side length of different shapes. This table could be used, for example, to determine the size of a hayloft needed to store a certain amount of hay.  

While this method worked fine, it was not a general solution. The next approach may have come from the Babylonians, who had an advantage over the Egyptians in that their number system was more like the one we use today (although it was hexagesimal, or base-60). This made addition and multiplication easier. It is thought that by around 400 BC, the Babylonians had developed the method of completing the square to solve generic problems involving areas. A similar method also appears in Chinese documents at around the same time.

The completing square method allowed the Babylonians and Chinese to cross-check the area values that they calculated for different purposes. 

The first attempts to find a more general formula for solving quadratic equations may have been made by Greek philosophers Pythagoras (c. 500 BC) and Euclid (c. 300 BC), who both used a geometric approach to deducing a general procedure for solving the quadratic equation.

Pythagoras observed that the value of a square root value is not always an integer. However, he refused to allow for proportions that were not rational. Euclid, in his mathematical treatise Elements, proposed that irrational square roots are also possible. 

However, because the ancient Greeks did not use the same number system that we now use, it was not possible to calculate the square root by hand, which is what architects and engineers really needed.

It was the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic equation, in his 628 AD treatise Brāhmasphuṭasiddhānta ('Correctly Established Doctrine of Brahma').

Indian mathematics used the decimal system. It also had one other advantage over the system used by the ancient Egyptians and Greeks — the zero. Zero allowed mathematicians to not only theorize about irrational numbers but to use them in equations.

Brahmagupta recognized that there are two roots in the solution to a quadratic equation and described the quadratic formula as, "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." This can be written as:

x = [√(4ac+b2) - b]/2a

This was also one of the first works to describe concrete ways of using zero. In the years that followed, the Indian astronomer Bhāskara mathematically confirmed the possibility that any positive number has two square roots.

Around 820 AD, Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī, who was familiar with the zero used in Indian mathematics, developed what we now know as algebra. He solved the quadratic equation using algebraic expressions (although he rejected negative solutions) and is often credited as the father of algebra. His work made its way to Europe by around 1100 AD, where it was translated into Latin.

By 1545, Italian scientist Gerolamo Cardano had compiled works related to the quadratic equations, including both Al-Khwarizmi's solution and Euclidean geometry. In his works, he allows for the existence of roots of negative numbers.

Flemish engineer and physicist Simon Stevin gave the general solution of the quadratic equation for all cases in his book Arithmetic in the year 1594. Later, French scientist René Descartes published the special cases of the quadratic formula in his 1637 work La Géométrie, which also used the mathematical notation and symbolism that had been developed by mathematician François Viète. Descartes's work included the quadratic formula in the form we know today.

The quadratic equation in real life

Quadratic equation came into existence because of the simple need to conveniently find the area of squared and rectangular bodies, but from the days of its origin, this popular maths equation has now come a long way to prove its significance in the real world.

  • Sports analysts and team selectors use different quadratic equations to analyze the performance of athletes over a period of time. Moreover, sporting events such as javelin and basketball use quadratic formulas to find the accurate distance, speed, or time required to score more.  
  • Military and law enforcement units use quadratic formulas to calculate the speed of missiles, moving vehicles, and aircraft. The landing coordinates of planes, tanks, and jets are also determined using the formulas from quadratic equations.
  • Auto parts such as brakes and curved elements are designed on the basis of the quadratic formula. Pension plans, insurance models, employee work performance; all these parameters are calculated using quadratic equations. Apart from these, the boundaries in agricultural lands and the area of fields with the highest yield are also measured by the means of the quadratic formula.
  • The construction of monuments, offices, flats, roads, bridges, and more involves complex calculations and area measurements, so all these mathematical complications are dealt with using different quadratic formulas.  
  • The angles at which a satellite dish is set to catch the signals are also determined using the quadratic equations. Also, to figure out the way a dish receives signals from multiple satellites at the same time, a quadratic equation is taken into account.

The quadratic formula is among the fundamental principles of modern-day mathematics. Every future engineer, scientist, or mathematician is destined to face the quadratic equation in one or the other form. 

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