In this chapter we have studied the equation #ax^2+bx+c=0# with unknown #x#. If #a=0# and #b\ne0#, the equation is linear. This case is dealt with in an earlier chapter.

In general when the linear equation has a single solution it is possible that the quadratic equation has two solutions.

Also, we have dealt with variations, such as #x^2+b\cdot |x| +c = 0# (here, #|x|# is the absolute value of #x#) and #ax^4+bx^2 +c = 0#.

The contents from the previous chapters could give the impression that any equation of the form #a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1 x+a_0=0# can be solved by a general formula such as the abc-formula for the quadratic case. This is not true. Especially for #n\ge5# such formulas do not exist.

Systems of nonlinear equations in several unknowns are also difficult to solve: this is due to the fact that methods are inefficient and because only few formulas are available.

The quadratic equation #x^2+1=0# has no real solutions. But if you allow two fictional (mostly called imaginary) solutions, you enter the world of complex numbers. A special feature of complex numbers is that every quadratic equation has two solutions (counting double the only solution if the left-hand side is a square).

An interpretation of the equations in this section in terms of features and graphs you can find in the chapter Functions of the Calculus course. This is because solutions of equation #ax^2+bx+c=0# of unknown #x# are the #x# coordinates of the solutions to the system of equations \[\eqs{y&=&ax^2+bx+c\cr y&=&0\cr}\] with unknowns #x# and #y#. Here #ax^2+bx+c# is the function definition of a function and #y# is the value of that function at the point #x#.

On the other hand, the equation #y=ax^2+bx+c# is a quadratic equation with two unknowns, which is dealt with geometrically in the chapter 2-dimensional geometry: conic sections. Before starting this chapter, we recommend you study 2-dimensional geometry : points and lines.