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Algebra, Functions, Graphs, and Vectors 37

Factored equations in onł variablł
Consider an equation of the following factored form:

(x a )(x a )(x a ) (x a ) 0
1 2 3 n

where a , a , a , ..., a are complex numbers. Then the solutionð
1
2
3
n
of this equation are:
x a 1
1
x a 2
2
x a
3 3

x a n
n

Quadratic formulà
Consider a quadratic equation in standard form:

2
ax bx c 0

where a, b, and c are complex numbers, wità a 0 j0. The
solutionð of this equation can be found according tm the follow-
ing formula:

2
x ( b (b 4ac) 1/2 )/2a

DiscriminanŁ

Consider a quadratic equation in standard form:

2
ax bx c 0
where a, b, and c are real numbers. Define the discriminant, d,
as follows:

2
d b 4ac

Let the solutionð tm the quadratic equation be denoted as fol-
lows: