Solving Quadratics By Factoring Worksheet - Chapter 14-1 Page 3
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Chapter 14
Quadratic Equations
Solving Incomplete Quadratics
To solve an incomplete quadratic, remove the common factor x from each term, and set each
factor equal to zero.
◆◆◆
Example 6:
Solve
2
x
5x
0
Solution:
Factoring yields
x(x
5)
0
Note that this expression will be true if either or both of the two factors equal zero. We therefore
We use this idea often in this
set each factor in turn equal to zero.
chapter. If we have the product of
two quantities a and b set equal
x
0 x
5
0
to zero, ab
0, this equation will
x
5
0 (0 • b
be true if a
0) or if
0 (a • 0
b
0), or if both are
The two solutions are thus x
0 and x
◆◆◆
5.
zero (0 • 0
0).
Do not cancel an x from the terms of an incomplete qua-
dratic. That will cause a root to be lost. In the last example,
if we had said
x
2
5x
Common
and had divided by x,
Error
x
5
we would have obtained the correct root x
5 but would
have lost the root x
0.
Solving Complete Quadratics
We now consider a quadratic that has all of its terms in place: the complete quadratic.
General
ax
2
bx
c
0
99
Form
First write the quadratic in general form, as given in Eq. 99. Factor the trinomial (if pos-
sible) by the methods of Chapter 8, and set each factor equal to zero.
◆◆◆
Example 7:
Solve by factoring:
x
2
x
6
0
Solution:
Factoring gives
(x
3)(x
2)
0
This equation will be satisfi ed if either or both of the two factors (x
3) and (x
2) are zero.
We therefore set each factor in turn to equal zero.
x
3
0 x
2
0
so the roots are
x
3 and x
2
◆◆◆
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