18 1. REVIEW OF ALGEBRA
Problem 1.43 Suppose we are tracking a point on a spinning disk over time. Find some com-
putable, linear quantity that describes the point’s position.
1.3 QUADRATIC EQUATIONS
After a line, the simplest type of equation is a quadratic equation. Its name comes from the
Latin word for “square,” and it is characterized by always including a squared term. When
graphed, it has a shape like that shown in Figure 1.5. is shape is called a parabola.
Knowledge Box 1.8
A quadratic equation is an equation in the form
y D ax
2
C bx C c
where a, b, and c are constants and a ¤ 0.
e usual goal, when we have a quadratic equation, is to find the values that make the
equation zero or satisfy the equation. If we have the equation y D x
2
3x C 2 and if x D 1,
we get 1 3 C 2 D 0, and so x D 1 satisfies the equation. Similarly, if x D 2, then we get
4
6
C
2
D
0
, and so
x
D
2
also satisfies the equation. A number that satisfies an equation is
a point where the graph of the equation crosses the x-axis. A solution to an equation is also
called a root of the equation. How can a quadratic equation have two solutions?
It turns out that a quadratic equation can have 0, 1, or 2 roots. Figure 1.6 shows how this is
possible.
When the graph of the equation never crosses the x-axis, there are no roots; when it just touches
the x-axis at one point, there is one root. When it crosses the x-axis twice, there are two roots.
ere is a formula we can use to tell how many roots a quadratic equation has. It is called the
discriminant.
Knowledge Box 1.9
e number of roots of a x
2
C bx C c D 0 is determined by looking at
the discriminant:
D D b
2
4ac
If D < 0, then the equation has no roots; if D D 0, then the equation
has one root; if D > 0, then the equation has two roots.
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