Quadratics

Completing the Square

Writing the standard form of a quadratic ax2+bx+cax^2+bx+c in vertex form a(x−h)2+ka(x-h)^2+k is called completing the square.

Procedure

  1. Start with the quadratic expression ax2+bx+cax^2+bx+c
  2. Factor out the coefficient of x2x^2 from the whole expression
  3. Complete the square by adding and subtracting (b2a)2\left(\frac{b}{2a}\right)^2
  4. Factorise the perfect square trinomial
  5. Simplify the expression
ax2+bx+caa(ax2+bx+c)a(x+bax+ca)a(x+bax+(b2a)2−(b2a)2+ca)a(x+bax+(b2a)2)−a((b2a)2+ca)a(x+b2a)2−b24a+cax^2+bx+c\\\frac{a}{a}\left(ax^2+bx+c\right)\\a\left(x+\frac{b}{a}x+\frac{c}{a}\right)\\a\left(x+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2-\left(\frac{b}{2a}\right)^2+\frac{c}{a}\right)\\a\left(x+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2\right)-a\left(\left(\frac{b}{2a}\right)^2+\frac{c}{a}\right)\\a\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a}+c

Vertex Form

Completing the square allows the identification of the vertex (h,k)\left(h,k\right) of a parabola. The vertex is the turning point and lies on the axis of symmetry.

h=−b2ak=c−b24ah=-\frac{b}{2a}\\k=c-\frac{b^2}{4a}