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
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!