# Useful Formulae

16 THEQUADRATICFORMULA discriminants and parabolas A quadratic equationis one of the form ax2 bx c 0 where a is not zero. The solutions,or roots,of such an equation are given by the quadratic formula The quantity b2 4acis called the discriminant which dictates the nature and number of the solutions. There are three cases b2 4ac0 two different real solutions b2 4ac 0 only one real solution b2 4ac0 two different complex orimaginary solutions as opposed to real ones see page 50 Examples shown opposite i 2x2 x 1 0 has a discriminant of 9 and produces two real solutions,1 and . ii x22x1 0 yields a discriminant of zero and hence has one real root,namely x 1. iii 4x2 8x 5 0 is an example of a quadratic equation with no real roots,solving for x 1 and x 1 see page 50. The quadratic equation ax2bx c 0 has real solutions at values of xwhere the graph of the function y ax2 bx ccrosses the xaxis i.e. where y 0. 2i 2i b b2 4ac 2a 17