For those of you who know me personally, you’ll know that I have been tutoring maths for six years – GCSE and A-Level. Some people my age, or even just those past GCSE level, tell me they can’t remember how to do some things – so I’m hoping to use REFRESHERS to renew your memories – or to teach you something new!

**What is a quadratic equation?**

A quadratic equation is any equation that can be re-arranged to look like this…

aχ² + bχ + c = 0

…with a, b and c being constants (numbers). For example:

χ² + 5χ – 14 = 0

NB: In this example ‘a’ = 1.

**What do we mean by ‘solve’?**

What we are looking for here is to find the unknown. In the above case the unknown is *x*, but it could represented by any letter – so watch out for that.

Quadratics will have two solutions. Why? Because of the “SQUARED” on the *ax*, it means that when plotted on a graph, the equation crosses the *x *axis twice. Like this:

**How do you solve them?**

There are three ways to solve quadratic equations. Factorising, Completing the square and the quadratic formula.

**Factorising – the easy and quick way**

Factorising involves taking the equation and putting it into two brackets. that look like this:

(*x* …. )( *x* …. ) = 0

Let’s use the above example of:

χ² + 5χ – 14 = 0

To factorise you have to find two numbers that MULTIPLY to make *c* (-14) and ADD to make *b* (5). [Watch out for minus numbers – this can trip you up!] The solution here is:

(χ + 7)(χ – 2) = 0

Seven and negative two **multiply to make -14 but add to get 5**. But how does this help us solve the equation? See the = 0? It means that each bracket must answer 0. How do you make 7 equal 0? By taking away 7. And how do you make -2 equal 0? Add 2. So..

χ = -7 and x = 2

2. **Completing the square – the more complicated way **

Let’s use another example for this one:

χ² + 10χ + 2 = 0

Completing the square involves rearranging your formula and finding what I call ‘the magic number’. When *a* = 1, rearrange your formula to look like this (ignore c for a second):

(χ + b ∕ 2)²

In our example, this will become (because b = 10):

(χ + 5)²

From here you can find the “magic number”. Expand the brackets. In this case the expansion comes to:

(χ + 5)² = χ² + 10χ + 25

But that doesn’t like quite like our original formula – it’s got an extra 25 (the magic number!) so we can TAKE AWAY the 25, but remember to ADD BACK ON *c* (in this case, 2). Thus:

(χ + 5)² – 25 + 2

= (χ + 5)² – 23

But how does this help us solve the equation? Remember that this equation = 0. So now we can do some rearranging to find x.

(χ + 5)² = 23

χ + 5 = √23

χ = – 5 ± √23

We can either leave it here, or plug it into our calculator twice, one with – and once with + in the place of ±. This comes out as:

χ = −0.2041684767 and −9.795831523

3. **The Quadratic Formula – the last resort **

Finally, we have the final way, only to be used if: a) the other two options don’t work or b) the question asks you to give your answer to 2 d.p. (decimal places). Simply (she says) plug a, b and c into the following formula:

In the case of:

χ² + χ + 7 = 0

We would find that: a = 1, b = 1 and c = 7. Plugged in this looks like:

(- 1 ± √1² – 4 x 1 x 7) ∕ 2 x 1

Once you’ve plugged this all in you can use your calculator to find the two solutions, one with – and once with + in the place of ±.

χ = (-1 + √1² – 4 x 1 x 6) / 2 x1

and

χ = (-1 – √1² – 4 x 1 x 6) / 2 x1

**Try these yourself! **

- Factorise: χ² – 2χ – 15 = 0
- Complete the Square: χ² + 6χ + 1 = 0
- Use the Quadratic Formula: 4χ² – 7χ + 3 = 0

**A final note**

When factorising or completing the square, quadratics become more complex when a ≠ 1 (a does not equal one). I will cover further examples another time!