Beautiful Equations Explained… the Quadratic Formula

The Quadratic formula, the ABC (and x) of algebra! However, while you keep using the alphabet past high school, solving quadratic equations comes up a bit less in day to day life and so you’d be forgiven for having forgotten exactly what this beauty of a formula is for. Luckily for you, we’re passionate about sharing beautiful mathematics with the world! Here’s a quick recap on how this equation works, and why it’s so useful!

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The quadratic formula provides the value(s) for x, at which the quadratic function ax2 + bx + c where a, b and c are constants, is equal to zero.

Or in other words: say you have an equation made up of “x squared”, “x” and some other number and all these parts are added or subtracted together. You’re told the equation equals zero and you need to find out the value of this x. You’ve got a few options: factorise the equation or “complete the square” for example, but if you just can’t seem to find the right numbers to do so, then this formula is exactly what you need! By taking the number in front of “x squared” and subbing it in for “a” in the equation, the number in front of “x” for “b” and the number on its own for “c”, then just like that, you’re finished! You’ve found the values of “x” that when put back into your original equation will give you 0.

Graphically this formula represents the values for x for which the function is equal to zero, hence the points at which it crosses the x-axis. Quadratic functions are found as commonly in real life systems as they are in algebra: in geometry determining the radius r of a circle given it has area A, modelling the trajectory of the tennis ball being thrown in mechanics, or calculating profits in finance.

There is evidence of Geometric methods being used to solve quadratic equations as far back as 2000 BC in Babylonia, Egypt, Greece, China, and India. The famous Greek mathematicians Euclid and Pythagoras are credited with the first generalised geometric method; however, it is thought to be the Indian mathematician and astronomer Brahmagupta who gave the first explicit formula for the solutions in 7th century AD. The philosopher René Descartes, best known for the philosophical statement “I think, therefore I am”, published La Géométrie in 1637, which contained the first quadratic formula in the form we know today.