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### One dimension: Points

A point is specified by a single real number called its
*coordinate.* Let the coordinate of P_{1} be
x_{1}, and that of P_{2} be x_{2}.
The distance from P_{1} to P_{2} is

d = sqrt[(x_{1}-x_{2})^{2}] = |x_{1}-x_{2}|.

The coordinate of the point dividing the line segment
P_{1}P_{2} in the ratio r/s is
[rx_{2}+sx_{1}]/[r+s]. As a special case, when r = s,
the midpoint of the line segment has coordinate
[x_{2}+x_{1}]/2.

The set of all points with coordinate x satisfying a linear equation in
x is a single point. Its equation has the general form

Ax + B = 0,

where A is nonzero. The coordinate of the point is -B/A.

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### One dimension: General Quadratic Equations

A general quadratic equation can be put into the following form:

ax^{2} + 2bx + c = 0,

where a is nonzero. If we have an equation of this kind, it can
represent one of three different kinds of curve. Which kind depends on
the value of the following quantity:

The cases are as follows:

Case sign(Delta) Name Standard form
1 - Real circle x^{2}/r^{2} = 1
2 + Imaginary circle x^{2}/r^{2} = -1
3 0 Coincident lines x^{2} = 0

The equation can be put into standard form by completing the square,
and then making a translation to move the center to the origin:

0 = ax^{2} + 2bx + c,
0 = a^{2}x^{2} + 2abx + ac,
0 = (ax + b)^{2} - b^{2} + ac,
-Delta = (ax + b)^{2},
-Delta/a^{2} = y^{2},

where y = x + b/a,

-sign(Delta) = y^{2}/r^{2},

where r = sqrt(|Delta|)/a, provided Delta is nonzero. The center of
the circle has coordinate -b/a, and the radius is r.

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### One dimension: Circles

A circle is the set of all points at a distance (or radius) r > 0 from
the center P_{1}. Its equation has the form
(x-x_{1})^{2} = r^{2},
|x-x_{1}| = r.

It consists of just two points, whose coordinates are x_{1} + r
and x_{1} - r.