See: List of castles Castles in England
Angus
Argyll and Bute
Dumfries and Galloway
Dundee
Edinburgh
Fife
Highland
Moray
Perthshire
Scottish Borders
Stirling
... parabola described by the quadratic equation touches the x-axis in a single point.) If the discriminant is positive, then there are two different solutions x, both of ...