Quadratic equations

• A quadratic equation is an equation which contains only one variable but that variable must only be raised to powers that are positive whole numbers with a maximum value of \(2\), that is, of the form:
  \[y=ax^2+bx+c\]
• Quadratic functions can be represented graphically by a curve called a parabola.
  
  
  
  
• The parabola can take two forms:
  
  
  1. a minimum turning point when (\(a>0\))
     
     
  2. a maximum turning point when (\(a<0\))
     
     
     
• In order to sketch a parabola you need to know:
  
  
  1. where it cuts the \(y\)-axis (\(y\)- intercept, i.e. \(x=0\))
     
     
  2. where it cuts the \(x\)-axis (if at all, i.e. \(y=0\))
     
     
  3. the coordinates of the turning point (axis of symmetry can be found \(x=-\frac{b}{2a}\)).
     
     
• There are two techniques available to solve quadratic equations:
  
  
  1. Factorisation works on the principle that if the product of two expressions is zero then one or both of those expressions must be zero.
  If \[(x-a)(x-b) = 0\] then \[x-a=0 \quad \mbox{or} \quad x-b=0\] giving \[x=a \quad \mbox{or} \quad x=b\]
  2. Quadratic formula.