What are the different methods of solving quadratic equations?

For your maths GCSE it is important that you understand the three main methods of solving quadtratics: factorisation, completing the square, and using the quadratic formula.

1. Factorisation:

The first step for factorisation is to see if a common factor can be taken out, this is the easiest way of solving a quadratic.

For example:
2x² + 4x = 0
In the case a factor of 2x can be taken out, making the equation look like this:
2x(x+2) = 0 
This would then be solved by setting each part equal to zero,
2x = 0 and x+2=0.
Rearranging these equations gives us the final solutions of 
x = 0 and x = -2.

If a common factor cannot be found, the next step is to try and put the equation into two brackets that are multiplied together.

For example:
x²+5x+6=0
Would be rewritten as:
(x+2)(x+3)=0
^^^ when these brackets are multiplied out they give the original equation. 

So in order to split the equation into two brackets we have to know which numbers are needed. The solution will be of the form
(ax+b)(cx+d) = 0, where a,b,c and d are integers. 

So in our example,
a * c must equal 1 to give us the original 1x².
a * d + b * c must be equal to 5 to give us 5x.
And b * d must be equal to 6 to give us our constant. 

2. Completing the square:

To 'complete the square' of a quadratic, the initial equation is rewritten as a (x + constant) bracket squared minus another constant to give the same value as the starting equation. This is easier shown than explained with words.

For example:

x²+10x+20 = 0
First, the coefficient of x (the ten infront of the x) is halved, and this is the constant used in the bracket with x.
This gives us:
(x+5)²  
But we want (x+5)² + a constant to be equal to x²+10x+30 = 0.

If we expand the squared bracket we get the x² and the 10x that we need, but we get a +25, when we need +20.

To fix this we just take off another 5 after our squared bracket giving us a final equation of 
(x+5)²  - 5 = 0  

To solve for x we just add 5 to both sides and take the square root. 
(x+5)²  = 5
x + 5 = +/- sqrt(5)
x = - 5 +/- sqrt(5)

3. Quadratic formula:

The last way of solving a quadratic is using the quadratic formula. 

In the following example a, b, and c represent the integers in front of each part of the quadratic.

For example:

ax² + bx + c = 0 

To solve this using the quadratic formula, the integers just have to be subbed into the following equation:

x = [-b +/- sqrt (b² - 4ac)] / 2a

This is quite difficult to type out but easy to actually use.

End:
I hope this step by step guide of the methods of solving quadratic equations has been useful!