A-Level MathsQuadratics

Solving Quadratics

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

  1. By factorising – this is the simplest method provided that the quadratic can be factorised.                           Example: 2x^2-5x-3=0\Rightarrow(2x+1)(x-3)=0\Rightarrow x=-\frac{1}{2}, x=3.
  2. Using the 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.
    The quadratic formula says that if ax^2+bx+c=0 then the roots are given by x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.
  3. Completing the squarecompleting the square this is another infallible method for finding roots if a quadratic can be solved.   Examplex^2+6x+5=0\Rightarrow(x+3)^2-4=0\Rightarrow x+3=\pm 2\Rightarrow 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).