The Quadratic Formula (Problem)

This problem is something that is pretty well known from our high school education – the quadratic formula. In a follow-up post, I will present the solution through a shortened version of the process by which a mathematician might have gone about finding a solution.

Equations are among the most important objects in mathematics. In algebra especially, solving equations is a central task. We solve equations for many reasons – they can help us deal with our finances, they tell us through many scientific laws how the world around us behaves, and they can help us predict certain future outcomes. It should not be surprising then that one of the broadest and most important questions in mathematics regards asking “what kinds of equations can we figure out how to solve”?

The first equations we learn how to solve in algebra class are one-variable equations. These are equations like $3x + 5 = 2$ , or more broadly $Ax + B = C$ for any numbers $A, B, C$ . We learn early on in algebra how to solve these. For the equation $3x + 5 = 2$ , we first subtract 5 from both sides and we end up with $3x = -3$ . We then divide by both sides by 3 and conclude that $x = -1$ . If you use the same process on $Ax + B = C$ , then we can conclude that $x = \dfrac{C-B}{A}.$ This is a complete solution to this type of equation.

The next most challenging type of equation are called quadratic. These are equations that involve regular numbers, $x$ , and $x^2$ , which is $x * x$ . For example, the equation $x^2 + 3x + 7 = 0$ is a quadratic equation. Before we move on, it is natural to ask why quadratic equations would matter, and why we would try to solve these first instead of some other type of equation. The reason that quadratic equations are the “next easiest” to solve is because they involve nothing more difficult than multiplying $x$ by itself – and only once. One very good reason to care about these is because the shape of graphs you get from quadratic equations, called parabolas, are exactly the shape that objects make when thrown through the air. Because Newton’s equation for gravity is essentially quadratic, this means that by understanding quadratic equations, we can understand how projectiles move through the air.

With equations like $Ax + B = C$ , we developed an overall solution of $x = \dfrac{C - B}{A}$ that will always work, no matter the values of $A, B$ or $C$ (so long as $A$ is not zero). The question of the quadratic formula is to try to find a similar solution to the quadratic equation

$ax^2 + bx + c = 0.$

As an example, we could as what all the solutions to $x^2 = 4$ are, and intuitively we can arrive at the answer of $x = 2$ and $x = -2$. But the equation $x^2 = 0$ has only one solution, $x = 0$ . We learn from this that we are sometimes looking for two solutions – not necessarily just one – and so we want to make sure our quadratic formula will tell us about both of these solutions, if there are in fact more than one.

This is about all you need to understand what our goal is – to find a systematic way to use the numbers $a, b$ , and $c$ to find solutions $x$ to the quadratic equation $ax^2 + bx + c = 0$ . In an upcoming post, we will show how this problem can be solved.