Algebra Rules Review Sheet Page 5
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REVIEW OF ALGEBRA
5
EXAMPLE 10
2
2
2
2x
12x
11
2 x
6x
11
2 x
6x
9
9
11
2
2
2 x
3
9
11
2 x
3
7
QUADRATIC FORMULA
By completing the square as above we can obtain the following formula for the roots of a
quadratic equation.
2
7
The Quadratic Formula
The roots of the quadratic equation
ax
bx
c
0
are
2
b
4ac
sb
x
2a
2
EXAMPLE 11
Solve the equation
5x
3x
3
0
.
SOLUTION
With
a
5
,
b
3
,
c
3
, the quadratic formula gives the solutions
2
3
s3
4 5
3
3
s69
x
2 5
10
2
The quantity
b
4ac
that appears in the quadratic formula is called the discriminant.
There are three possibilities:
2
1.
If
b
4ac
0
, the equation has two real roots.
2
2.
If
b
4ac
0
, the roots are equal.
2
If
b
4ac
0
, the equation has no real root. (The roots are complex.)
3.
These three cases correspond to the fact that the number of times the parabola
2
y
ax
bx
c
crosses the -axis is 2, 1, or 0 (see Figure 1). In case (3) the quadratic
x
2
ax
bx
c
can’t be factored and is called irreducible.
y
y
y
0
0
0
x
x
x
FIGURE 1
(a) b@-4ac>0
(b) b@-4ac=0
(c) b@-4ac<0
Possible graphs of y=ax@+bx+c
2
EXAMPLE 12
The quadratic
x
x
2
is irreducible because its discriminant is negative:
2
2
b
4ac
1
4 1 2
7
0
2
Therefore, it is impossible to factor
x
x
2
.
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