Morning Mathematical Monsters & Maniacs!
(Today’s post is sponsored by the letter “M”)
Over the past 600+ episodes, The Simpsons has taken us on an amazing mathematical journey involving fractions, probability, Fermat’s last theorem, and hundreds of other aspects from the wonderful world off mathematics.
And what better way to start your week, then by discussing math Monday morning?
This week, we’re going to look at one of algebras most famous equations and solve it – as we discuss the quadratic equation and quadratic formula.
In Season 5, Episode 9 The Last Temptation of Homer, while Homer is busy dealing with his new co-worker Mindy, Dr. Hibbert tells Marge that Bart has a lazy eye and many other disorders which is why he is an underachiever. As a result of addressing Bart’s array of maladies, he is turned into a nerd, with orthopedic shoes and thick glasses. Bart then becomes a stereotypical nerd and is picked on by the bullies. Martin welcomes and shows Bart the hideout where all nerds go to in order to escape bullies. It is in this hideout that we come across the quadratic formula being written on the wall.
In algebra, a quadratic equation is any equation having the form
ax2 + bx + c = 0,
where x represents an unknown; and a, b, and c represent known numbers known as coefficients. Specifically, a is known as the quadratic coefficient, b as the linear coefficient, and c as the constant or free term. The values of x that satisfy the equation are called solutions, roots of zeros, or the left-hand side.
In the quadratic equation, a cannot equal 0. If a = 0, then the equation is linear, not quadratic, as there is no ax2 term. We’ll discuss linear equations at a later date.
A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real. The solution to the quadratic equation can be solved in a variety of ways, including methods such as factoring, completing the square, quadratic factorizations, geometric interpretations, etc. The solution of the quadratic equation is known as the quadratic formula.
Regardless of the method used to solve for it, the quadratic equation can be solved to result in the quadratic formula having the form
x = [ -b ± √(b2 – 4ac) ] / 2a
The quadratic equation and quadratic formula are named for the latin of square. Solutions to the quadratic equation date back to 2,000 BC, to the Babylonian third dynasty of Ur. Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India over 4,000 years ago.
Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach, Greek mathematicians Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root. In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit solution of the quadratic equation. 9th century Persian mathematician Muhammad ibn Musa al-Khwarizmi developed a set of formulas that worked for two positive solutions, as well described the method of completing the squares. 10th century Egyptian mathematician Abū Kāmil Shujā ibn Aslam was the first to accept irrational numbers as solutions. In 1637, René Descartes (pictured below) published La Géométrie containing the quadratic formula in the form we know today.
Now we know more about one of the most famous algebraic equations and formulas. Were you familiar with the quadratic formula? Did you notice it on the walls in the episode? Sound off in the comments below. You know we love hearing from you.