In linear equations with unknown #x#, terms occur that are constant or contain #x#. In quadratic equations also terms containing #x^2# occur. 

If you want a rectangular area to be exactly #100\ {\rm m}^2# (square meters) and the width #x# to be exactly #10\ {\rm m}# less than the length, you must ensure that #x\cdot (x+10) = 100#. By expanding all the brackets and moving all the terms to the left hand side, we get the quadratic equation #x^2+10x-100 = 0#. We will see that the solution to this equation is equal to #x =-5-\sqrt{5} \lor x= -5+\sqrt{5}#. As the first solution is negative, the width should be #-5+\sqrt{5}\ {\rm m}#. Consequently, the dimensions of the requested rectangle are #6.18\times 16.18\ {\rm m\times m}#.

If we would have chosen #200# as the desired area instead of #100#, no root would have occured in the solution ( #10\times 20\ {\rm m\times m}# ). Generally speaking, a single root is needed to describe the solution of the equation. In the course of this chapter we will show you how you can use the famous quadratic formula to find our if this is the case or not.

As with Linear equations with one unknown, we will briefly discuss several variations.