Solving Quadratics

There are three ways in which you can solve quadratics – each method requires setting the quadratic to 0 first:


This is the simplest method provided that the quadratic can be factorised.


2x^2-5x-3=0\hspace{5pt}\Rightarrow\hspace{5pt}(2x+1)(x-3)=0\hspace{5pt}\Rightarrow\hspace{5pt} x=-\frac{1}{2}, x=3.

Quadratic Formula

If a quadratic cannot be factorised but does have roots, then the quadratic formula will find them. Recall that the discriminant will tell you how many roots a quadratic has.  See Discriminants page.
The quadratic formula says that if ax^2+bx+c=0 then the roots are given by:


Completing the Square

This is another infallible method for finding roots if a quadratic can be solved. See Completing the Square page.


x^2+6x+5=0\hspace{3pt}\hspace{5pt}\Rightarrow\hspace{5pt}(x+3)^2-4=0\hspace{5pt}\Rightarrow\hspace{5pt} x+3=\pm 2\hspace{3pt}\hspace{5pt}\Rightarrow\hspace{5pt} x=-1, x=-5

Sketching Quadratics

Completing the square is also useful for sketching a quadratic. The reason for this is that, by writing the quadratic in completed square form, we can see the transformations applied to the graph of x^2. For example, y=(x-3)^2+1 is the graph of x^2 shifted to the left by 3 (x transformation) and then up by 1 (y transformation).

1. Find the roots using one of the above methods, roots occur when y=0.

2.  Find the y-intercept, this occurs when x=0.

3. Find the coordinates of the vertex by completing the square and applying transformations to y=x^2.

A-Level Maths

Click here to find Questions by Topic – scroll down to all past QUADRATIC questions.

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