Quadratic Equations and the Quadratic Formula

Okay, first things first.  Since I’m aiming this blog to be accessible to all, this entry is going to be more aimed for those who haven’t studied Maths in a while (or anyone keen enough to be reading this whilst doing GCSEs!).  So if you don’t fall into those categories, feel free to gloss over this one!

 

Last time, I briefly mentioned things called “polynomials”; an equation in terms of powers of .  A quadratic equation is a polynomial that goes up to the term.  So, a quadratic equation looks like that:  , where a, b and c are just numbers.  You may ask yourself:  “What type of number?”, given that I explained the different types of numbers earlier.  Let’s just say that they’re a “real number”, so basically any number.  And, unless I say otherwise, we’ll always assume they are.  Simples.

 

One important thing with maths is being able to solve things.  So, we want to be able to write ‘what x is’ when .  This is where the ‘quadratic formula’ comes in.  The quadratic formula gives you solutions to any quadratic equation, just so long as you know what the numbers a, b and c are.  It’s actually quite simple to get to this formula, just from the equation.  However, since I want to avoid actually deriving equations and such, I’m not going to go over it here (plus, a WordPress blog isn’t the most suitable place!).   If you’re interested, I’d recommend this link – http://www.mathsisfun.com/algebra/quadratic-equation-derivation.html

 

So, we can now solve the quadratic equation.  All you have to do is plug in some numbers into a formula (and you can even use a calculator to do that, woo!).  However, such formula aren’t always possible.  Indeed, whilst there is a formula for solving cubic equations (polynomials up to a third power, so of the form ), there does not exist any formula for any other polynomials.  Even the ‘cubic formula’ is a very long and complicated (if you’re interesting, just search ‘Cubic formula’ online).  

 

As you can tell, such formula are very rare, making them powerful tools in maths.  So, we can’t rely on there always being a special formula; meaning maths is about learning the different methods and ways of solving things, trying these things and hoping they work.  And if they don’t, then you just try another one.  But let’s not worry about and just take comfort in knowing the quadratic formula is always there to save us…